Regular spiking in high conductance states: the essential role of inhibition
DDOI: 10.1103/PhysRevE.103.022408Published in Phys. Rev. E , 022408 (2021)
Regular spiking in high conductance states: the essential role of inhibition
Tomas Barta ∗ Institute of Physiology of the Czech Academy of Sciences, Prague, Czech RepublicCharles University, First Medical Faculty, Prague, Czech Republic andInstitute of Ecology and Environmental Sciences, INRA, Versailles, France
Lubomir Kostal † Institute of Physiology of the Czech Academy of Sciences, Prague, Czech Republic (Dated: February 19, 2021)Strong inhibitory input to neurons, which occurs in balanced states of neural networks, increasessynaptic current fluctuations. This has led to the assumption that inhibition contributes to thehigh spike-firing irregularity observed in vivo . We used single compartment neuronal models withtime-correlated (due to synaptic filtering) and state-dependent (due to reversal potentials) inputto demonstrate that inhibitory input acts to decrease membrane potential fluctuations, a resultthat cannot be achieved with simplified neural input models. To clarify the effects on spike-firingregularity, we used models with different spike-firing adaptation mechanisms and observed thatthe addition of inhibition increased firing regularity in models with dynamic firing thresholds anddecreased firing regularity if spike-firing adaptation was implemented through ionic currents or notat all. This novel fluctuation-stabilization mechanism provides a new perspective on the importanceof strong inhibitory inputs observed in balanced states of neural networks and highlights the keyroles of biologically plausible inputs and specific adaptation mechanisms in neuronal modeling.
INTRODUCTION
In awake animals, neocortical neurons receive a streamof random synaptic inputs arising from background net-work activity [1–3]. This “synaptic noise” is respon-sible for the fluctuations in membrane potential andstochastic nature of spike-firing times [4–10]. Since spike-firing times encode the information transmitted by neu-rons, investigating the properties of neuronal responsesto stochastic input, representing pre-synaptic spike ar-rivals, is of significant interest.Typically, the total conductance of inhibitory synapsesis several-fold higher than that of excitatory synapses[11]. This state, commonly referred to as the “high con-ductance state” has been demonstrated to significantlyaffect the integrative properties of neurons [2, 12–15].Concurrently, the high inhibition-to-excitation ratio in-troduces additional synaptic noise, which should intu-itively result in noisier firing. However, studies havedemonstrated that the high ratio of inhibition may leadto more efficient information transmission [16–18].
Invivo studies have also demonstrated that the onset ofstimuli can stabilize the membrane potential without asignificant change in its mean value [19, 20]. Monier et al.[19] observed that the decrease in fluctuations was asso-ciated with higher evoked inhibition, which may have ashunting effect [21]. Nevertheless, a theoretical frame-work explaining why and under which conditions thisshunting effect overpowers the increased synaptic noiseis lacking. ∗ [email protected] † [email protected] Synaptic input can be modelled as temporary openingof excitatory and inhibitory ion channels, which act toeither depolarize or hyperpolarize the neural membrane,respectively. Statistical measures of membrane potentialcan be calculated exactly with the resulting expressionsbeing non-analytic [22] or they can be approximated inthe steady-state with the effective time-constant approx-imation [23, 24]. For better analytical tractability, thesynaptic drive is often simplified with one (or both) ofthe following assumptions:A1 The magnitude of the synaptic current elicited byeach presynaptic spike is independent of the voltage[25–30], orA2 Time profiles of individual synapses (synaptic fil-tering) are neglected [8, 23, 30–34].In order to observe the shunting effect of inhibition[19], reversal potentials have to be considered, which ex-cludes assumption A1. Richardson [23] demonstratedthat an increase in inhibition could decrease the mem-brane potential for strongly hyperpolarized membranesin a model of synaptic input with omitted synaptic fil-tering (assumption A2). However, we demonstrate thatif neither of the simplifying assumptions are used, themembrane potential stabilization effect can be observedacross the complete range of membrane potentials, de-spite increased synaptic current fluctuations (Fig 1A,B).This naturally poses the question if the decreasedmembrane potential fluctuations lead to more regularfiring activity [20]. To this end, we analyze the effectof membrane potential stabilization on different neu-ronal models. In particular, we focus on how the ef-fects of inhibition change for different spike-firing adap-tation (SFA) mechanisms. SFA is responsible for the a r X i v : . [ q - b i o . N C ] F e b decrease of a neuron’s firing rate in response to a sus-tained stimulus and plays a crucial role in all stages ofsensory processing (e.g., [35–40]). We compare two dis-tinct SFA mechanisms: adaptation through ionic cur-rents (muscarinic currents, AHP currents) and adapta-tion through dynamic threshold. We demonstrate thatdespite their formal similarities [41, 42] the effect of in-hibition qualitatively differs for these SFA mechanisms(Fig 1C,D). We illustrate the differences on the analyti-cally more tractable generalized leaky integrate-and-firemodels (GLIF) followed by the biophysically more plau-sible Hodgkin-Huxley (HH)-type models. METHODSSubthreshold membrane potential
In order to analyze the behavior of neurons in the ab-sence of any spike-firing mechanism, we consider a pointneuronal model with membrane potential V described by C d V ( t )d t = − g L ( V ( t ) − E L ) + 1 a ( I e ( t ) + I i ( t )) , (1)where C is the specific capacitance of the membrane, g L isthe specific leak conductance, E L is the leakage potential, I e , i are the synaptic currents due to stimulation by affer-ent neurons through excitatory and inhibitory synapses,respectively, and a is the membrane area [43, 44]. Forbrevity, we will further use V ≡ V ( t ). The synaptic cur-rents are described by I e , i ( t ) = g e , i ( t )( V − E e , i ) , (2)where g e ( t ), g i ( t ) are the total excitatory and inhibitoryconductances, and E e , E i are the respective synaptic re-versal potentials.The total conductances in the Eq (2) are given by g e , i ( t ) = (cid:88) t k ∈T e , i h e , i ( t − t k ) , (3)where T e , i are sets of presynaptic spike times modeledas realizations of stochastic point processes and h e , i arefiltering functions (i.e., time profiles of individual excita-tory and inhibitory conductances).Unless stated otherwise, we used the following parame-ters: C = 1 µ F / cm , g L = 0 .
045 mS / cm , E L = −
80 mV, E e = 0 mV, E i = −
75 mV, a = 3 . × − cm [45]. Spike firing models
GLIF models
We consider three versions of the GLIF model:
TABLE I.
GLIF models parameters
LIF AHP-LIF DT-LIF θ , θ (mV) -50 -50 -50 τ AHP (ms) - 100 -∆ g AHP (nS) 0 5 0 E K (mV) - -100 - τ θ (ms) - - 100∆ θ (mV) 0 0 4
1. The classical Leaky Integrate-and-Fire model(LIF),2. LIF with SFA through ionic (after-hyperpolarization) currents (AHP-LIF),3. LIF with SFA through dynamic threshold (DT-LIF).The membrane potential of the LIF model obeys theEq (1). Whenever
V > θ , where θ is a fixed thresholdvalue, a spike is fired, and the membrane potential V isreset to a value V r . For our simulations, we used θ = −
55 mV and V r = E L .In the model with AHP current SFA (AHP-LIF), anadditional hyperpolarizing conductance g AHP is includedin the model, and the membrane potential then obeysthe equation [46–48]: C d V d t = − g L ( V − E L ) − g AHP ( t )( V − E K ) + I e ( t ) + I i ( t ) , (4)or equivalently τ AHPef d V d t = − ( V − V AHPef ) , (5) τ AHPef ( t ) = aCg e ( t ) + g i ( t ) + g AHP ( t ) + ag L , (6) V AHPef ( t ) = ag L E L + g e ( t ) E e + g i ( t ) E i + g AHP ( t ) E K g e ( t ) + g i ( t ) + g AHP ( t ) + ag L . (7)where E K is the potassium reversal potential, and g AHP is the corresponding conductance which increases by∆ g AHP when a spike is fired and otherwise decays expo-nentially to zero with a time constant τ AHP . V AHPef thenrepresents the effective reversal potential. Note that forsimplicity, we omitted the voltage dependence of g AHP .In the dynamic threshold model (DT-LIF), the thresh-old increases by ∆ θ after each spike and then decreasesexponentially to θ with time constant τ θ .The parameters for the GLIF models are specified inthe Tab. I. Hodgkin-Huxley models
We adopted HH-type models developed by Destexheet al. [49]. The membrane potential obeys the equation:
FIG. 1. A: During a 2 s long simulation, the intensity of inhibitory input increases from 0 kHz to 20 kHz. The pre-synapticspike trains are modeled as Poisson point processes. B: The intensity of excitatory input is increased simultaneously with theinhibition in order to maintain the mean membrane potential constant. This increases fluctuations of the synaptic current butdecreases fluctuations of the membrane potential. The orange lines signify the mean value ± standard deviation. C and D:The effect of membrane potential stabilization on firing regularity. The intensity of the inhibitory input follows the time courseshown in A, and the intensity of the excitatory activity is increased in order to maintain the steady state post-synaptic firingrate at approximately 10 Hz (blue trace). The firing regularity (measured here by the Fano factor, orange trace) decreases withthe addition of inhibition to the input in the model with spike-firing adaptation by M currents (C). However, the model withdynamic threshold (D) exhibits a clear increase in regularity with the added inhibitory input. The mean firing rate and Fanofactor were calculated by a sliding window of length 100 ms from approximately 10 trials. C d V d t = − g L ( V − E L ) − g Na m h ( V − E Na ) −− g K n ( V − E K ) − g M p ( V − E K ) − a I syn , (8)where E Na and E K are the sodium and potassium re-versal potentials, respectively; g Na , g K , and g M are peakconductances; and m , h , n , and p are gating variablesobeying the equation:d x d t = α x ( V )(1 − x ) − β x ( V ) x, (9)or equivalently: τ x ( V ) d x d t = − ( x − x ∞ ( V )) , (10)where x is the respective gating variable, α x and β x arethe activation and inactivation functions, respectively,and τ x ( V ) = 1 α x ( V ) + β x ( V ) , (11) x ∞ ( V ) = α x ( V ) α x ( V ) + β x ( V ) . (12) The activation and inactivation functions are definedas follows: α m = − . V − V T − − ( V − V T − / − , (13) β m = 0 . V − V T − V − V T − / − , (14) α h = A h exp( − ( V − V T − V S − / , (15) β h = 41 + exp( − ( V − V T − V S − / , (16) α n = − . V − V T − − ( V − V T − / − , (17) β n = 0 . − ( V − V T − / , (18) α p = 0 . V + 301 − exp( − ( V + 30) / , (19) β p = − . V + 301 − exp(( V + 30) / . (20)We set g M = 0 in both the HH-0 and HH-DT models,and g M > h , which is TABLE II.
Parameters of the HH models
HH-0 HH-M HH-DT g Na (mS / cm ) 50 50 50 g K (mS / cm ) 5 5 5 g M (mS / cm ) 0 0.5 0 E Na (mV) 50 50 50 E K (mV) -90 -90 -90 V T (mV) -58 -58 -58 V S (mV) -10 -10 14 A h (ms − ) 0.128 0.128 0.00128 responsible for deactivating voltage-gated sodium chan-nels after firing a spike, by changing the parameters A h and V S . For more details see the Supplementary Figure1. The parameters for the three HH-type models (with-out SFA (HH-0) / M-current SFA (HH-M) / dynamicthreshold SFA (HH-DT)) are specified in the Tab II. Simulation details
For synapses, we used the exponential filtering func-tion: h e , i ( t ) = (cid:40) A e , i exp( − t/τ e , i ) t ≥ t < A e = A i = 0 . µ S, τ e = 3 ms, τ e = 10 ms. Suchinput parameters with intensities λ e = 2 .
67 Hz and λ i =3 .
73 kHz provide an input with g = 12 nS, σ e = 3 nS, g = 57 nS, and σ i = 6 . V n +1 = ( V ef ) n +1 + (cid:0) V n − ( V ef ) n +1 (cid:1) exp (cid:18) ∆ tτ n +1 (cid:19) , (22) V ef = (cid:80) x ∈ X g x E x (cid:80) x ∈ X g x (23) τ ef = C (cid:80) x ∈ X g x (24)where X contains all the channel types (synaptic, leak,voltage-gated, and adaptive). The update rule for thesynaptic conductances g e , i was( g e , i ) n +1 = ( g e , i ) n exp (cid:18) ∆ tτ e , i (cid:19) + N e , i A e , i , (25)where ( N e , i ) is a Poisson random variable with mean λ e , i ∆ t .We used the step size ∆ t = 0 .
025 ms.
Evaluating firing rate regularity
A classical measure of the firing regularity of steadyspike trains is the coefficient of variation (C V ), defined as follows (e.g., [50]): C V = σ ISI µ ISI , (26)where µ ISI and σ ISI are the mean and standard deviationof the interspike intervals (ISIs), respectively. Lower C V indicates higher firing regularity.To achieve an accurate estimate of the C V , we es-timated the statistics from approximately 160,000 ISIsfor each data point. For a Poisson process (C V = 1)with this number of ISIs, the estimate of C V falls within[0 . , . V . RESULTSMembrane potential is stabilized with increasedinput fluctuations
Since the inputs to a neuron consist of pooled spiketrains from a large number of presynaptic neurons, ac-cording to the Palm-Khintchine theorem [51], it is suffi-cient to approximate the excitatory and inhibitory inputsby Poisson processes with intensities λ e and λ i , respec-tively [43]. It has been demonstrated that this condi-tion is not necessarily satisfied for neurons in vivo [52].However, as we discuss below, this should not affect theconclusions of our analysis. According to Campbell’s the-orem [53], it then holds for the mean g , i and variance σ , i of the input g , i = λ e , i (cid:90) ∞ h e , i ( t ) d t, (27) σ , i = λ e , i (cid:90) ∞ h , i ( t ) d t. (28)Therefore σ e , i g , i = O (cid:18) √ λ e , i (cid:19) (a well-known property ofthe Poisson shot noise [43]).For the purposes of our analysis, we consider thevoltage equations of a membrane without any spike-generating mechanism as: τ ef ( g e ( t ) , g i ( t )) d V d t = − V − V ef ( g e ( t ) , g i ( t )) , (29) τ ef ( g e , g i ) = aCag L + g e + g i , (30) V ef ( g e , g i ) = ag L E L + g e E e + g i E i g L + g e + g i . (31)For large inputs σ e , i g , i (cid:28)
1, we can linearize the Eq (31): V ef ( g e , g i ) . = E (cid:18) − g Fe + g Fi ag L + g + g (cid:19) + g Fe E e + g Fi E i ag L + g + g , (32)where E = V ef ( g , g ) and g Fe , i = g e , i − g , i . Sincethe fluctuating terms in Eq (32) disappear with grow-ing input, evaluating the limits with a fixed inhibition-to-excitation ratio c = g g leads to:lim λ e ,λ i →∞ E [V ef ] = V ∞ ( c ) ≡ E e + cE i c , (33)lim λ e ,λ i →∞ Var [V ef ] = 0 . (34)Var [V ef ] is an upper bound on the variance of V (it fol-lows from the Eq (29) that the membrane potential V is essentially a “low-pass filtered” effective reversal po-tential V ef ). Therefore, it also holds that lim λ e ,λ i →∞ (cid:104) V (cid:105) = V ∞ ( c ) ≡ E e + cE i c and lim λ e ,λ i →∞ σ V = 0. This can alsobe observed from the perturbative approach suggestedin [54] and further developed in [23, 24]. Therefore,any membrane potential between the reversal potentials E i , E e can be asymptotically reached with zero vari-ance, despite the variance of the total synaptic current I syn = I e + I i increasing. Note that the Poisson condi-tion can be relaxed, since it is sufficient for this resultthat σ e , i g , i → σ V ( (cid:104) V (cid:105) ; c ) be the function specifying the stan-dard deviation of the membrane potential with mean (cid:104) V (cid:105) , parametrized by c . It is a continuous function, with σ V ( E L ; c ) = σ V ( V ∞ ( c ); c ) = 0, otherwise σ V ( (cid:104) V (cid:105) ; c ) > c leads to higher V ∞ ( c ). Therefore, given c > c , there has to be an interval close to V ∞ ( c )where c results in lower membrane fluctuations. More-over, simulations indicate that this holds, even in non-limit regimes (Fig 2A, top panel). This result is rathercounter-intuitive, since with an increase in c , it is neces-sary to increase both λ e and λ i (if (cid:104) V (cid:105) > E i ), and thus si-multaneously increase synaptic current fluctuations (Fig2A, bottom panel) in order to keep the membrane po-tential constant. With our choice of parameters, lower c may also result in a slight decrease in membrane po-tential fluctuations. This is mainly due to the membranetime constant τ = Cg L . = 22 ms. The shorter the time con-stant, the closer V follows V ef , and the smaller the regionin which decreasing c leads to lower membrane potentialvariability (see the Supplementary Figure 2). Effects on firing regularity
The regularity of spike-firing is important for infor-mation transmission between neurons [55–58]. In theprevious section, we demonstrated that if appropriatesynaptic drive is used, higher inhibitory input rates (orequivalently higher inhibition-to-excitation ratio c ) leadto lower membrane potential fluctuations. In this sec-tion, we focus on the effects of inhibition on post-synapticfiring regularity, particularly on the regularity of a post-synaptic spike train with a fixed frequency evoked bydifferent stimuli with different levels of inhibition. Generalized Leaky Integrate-and-Fire models
For our analysis, it is essential to distinguish two dif-ferent input regimes: 1. Sub-threshold regime: E ≤ θ and 2. Supra-threshold regime: E > θ , where θ is thefiring threshold.In the sub-threshold regime, firing activity is drivenby fluctuations in the membrane potential. Therefore,increasing the input rates λ e , λ i and simultaneously keep-ing E constant leads to a decrease in firing rate due tosuppressed membrane potential fluctuations (note thatan analogous effect was described in the Hodgkin-Huxleymodel [59]). In order to maintain the post-synaptic firingrate (PSFR) constant while increasing the input rates, itis necessary to compensate for the decrease in fluctua-tions by increasing E . Therefore, it is not intuitivelyclear whether the decrease in membrane potential fluc-tuations will lead to an increase in firing regularity.In the supra-threshold regime, the firing activity isgiven by the driving force on the membrane potential( V − V ef ) /τ ef . Fluctuations in the interspike intervals arethen given mostly by the fluctuations of V ef . However,lower fluctuations of V ef are associated with lower τ ef andit is necessary to decrease E , if one wishes to decreasethe fluctuations of V ef and keep the firing rate constant atthe same time. Intuitively, the fluctuations of V ef will im-pact the firing regularity more, if the difference ( V − V ef )is lower. Therefore it is again unclear how the increasedsynaptic fluctuations affect the firing regularity.In general, we observe that in the suprathresholdregime, the C V of ISIs decreases with growing PSFR (Fig3A,D). Moreover, as we show in the Appendix B:lim λ e ,λ i →∞ C V = 0 (35)However, if the firing rate is held constant, the C V in-creases with growing c . Therefore, an increase in theinhibition-to-excitation ratio decreases firing regularity,despite the stabilizing effect on membrane potential.With high values of c , the C V grows locally with in-creasing firing rate. This is due to the fact that as E isvery close to the threshold and the membrane time con-stant (Eq (30)) is very low, the neuron fires very rapidly(bursts) when V ef (Eq (29)) exceeds the threshold but isotherwise silent.For the AHP-LIF model, no improvements are ob-served in the firing regularity with increasing c (Fig3B,E). At low firing frequencies, the C V of the AHP-LIF model is generally lower than that in the classicalLIF model. This is to be expected given the introduc-tion of negative correlations in subsequent ISIs [60–62].However, at higher firing rates, higher c actually leads toa higher C V than that observed in the LIF model. This isdue to the fact that in regimes where V ef is always abovethe threshold in the LIF model, the hyperpolarizing M-current drives the time-dependent effective reversal po-tential V AHPef (Eq (7)) closer to the threshold. This leadsto bursting, similar to that observed in the LIF model
FIG. 2.
Stabilization of the membrane potential.
A, top panel: Membrane potential fluctuations as a function of the meanmembrane potential for different values of c . The full lines represent data obtained from simulations with different excitatoryinput intensities λ e . The dotted lines represent the effective time-constant approximation (ETA, Appendix A). Bottom panel:The standard deviation σ I of the total synaptic current I syn = I e + I i . Note that σ V decreases with growing c even though σ I increases. B: Overview of σ V (color) for all achievable (cid:104) V (cid:105) ( x -axis) at given c ( y -axis). C = 1 µ F / cm and g L = 0 .
045 mS / cm ,approximation of σ V computed from the ETA. Heatmaps for different values of g L are provided in the Supplementary Figure2 and for different values of A i (Eq. (21)) in the Supplementary Figure 3. with E near threshold. This is illustrated in more detailin the Appendix B, where we also demonstrate that if V ∞ ( c ) > V thr , then C V →
0, similar to the LIF model.In the DT-LIF model, with the limit of infinite con-ductances, the membrane potential will reach V ∞ ( c ) im-mediately after a spike is fired. If V ∞ ( c ) ≥ θ , the neuronwill fire with exact ISIs T = τ θ log (cid:18) θV ∞ ( c ) − θ (cid:19) . (36)Therefore, any firing rate lower than (cid:16) τ θ log (cid:16) ∆ θV ∞ ( c ) − θ (cid:17)(cid:17) − can be asymptoticallyreached with C V = 0. Thus, firing regularity canalways be improved by increasing c , similar to thecase of membrane potential variability. However, veryhigh input intensities are necessary to observe suchregularization. Further, with biologically realistic inputintensities (excitatory input intensity up to 100 kHz),increased regularity with higher c is observed only forpost-synaptic firing rates below approximately 20 Hz(Fig 3C,F).Note that the structure of the contour plot in Fig 3Fis very similar to that in Fig 2B, i.e., approximately for c >
1, an increase in c stabilizes the membrane poten-tial and increases the spike-firing regularity. The oppo-site is observed for c <
1. Moreover, the structure ofthe heatmap changes accordingly if the membrane timeconstant is decreased by increasing g L (SupplementaryFigure 2) or if the inhibitory synaptic connections are strengthened (Supplementary Figure 3). Hodgkin-Huxley models
Generally, the behavior of the HH models is very sim-ilar to that of their GLIF counterparts (Fig 4). Similar“subthreshold” behavior is apparent - for high values of c , the firing rate starts dropping to zero with increasinginput intensity.Similarly to the GLIF models, no improvements areobserved with growing c for the HH-0 (Fig 4A,D) andHH-M (Fig 4B,E) models. For the HH-DT model, lowerC V of ISIs can always be achieved in the subthresholdregime, when the rate starts dropping back to zero dueto the strong input (Fig 4C,F).Increasing c in the HH-DT subthreshold regime de-creases the C V . However, it is important to note thatincreased c does not imply stronger inhibitory input inthis case. In fact, increasing the inhibitory input rate λ i is almost always beneficial for the spike-firing regularityin the HH-DT model, and this is also the case in theDT-LIF model (Fig 5). From this, we conclude that ifa neuron exhibits a dynamic threshold, a stimulus willproduce a more regular spike train if it elicits an increasein inhibitory input simultaneously with excitatory input. FIG. 3.
The effect of membrane potential stabilization on spiking regularity in the GLIF models.
A-C: Dependenceof the C V of ISIs on the post-synaptic firing rate for different values of c (color-coded). The dotted parts of the curves representthe sections where λ e >
100 kHz. In the LIF and AHP-LIF model, higher c universally leads to higher C V . In contrast, in theDT-LIF model, higher c can lead to more regular spike trains, especially if the input intensities are high. If V ∞ ( c ) ≤ θ (or θ for the DT-LIF model, i.e., c ≥ . V , c onthe y -axis. If more than one input can produce the same PSFR with the same c , the lowest possible value of C V is color-coded,resulting in the discontinuity in F. The data points were obtained from simulations with different input intensities λ e , λ i . DISCUSSIONSimplified input models
Absence of reversal potentials
If the reversal potentials are not taken into account,the synaptic currents are given by I e , i ( t ) = (cid:88) t k ∈T e , i H e , i ( t − t k ) , (37)where H is again a filtering function. If the two currentsare uncorrelated, they will add up to an input currentwith mean value I and standard deviation σ I . If the dif-fusion approximation is employed (the current is modeledas an Ornstein-Uhlenbeck process with a time constant τ I ), the mean and standard deviation of the membranepotential are [63]: (cid:104) V (cid:105) I = E L + I g L , (38) σ V,I = σ I τ I a g L ( C + g L τ I ) . (39) In the absence of synaptic reversal potentials, the vari-ance diverges with growing input, and increasing thesynaptic current fluctuations by increasing λ e and λ i clearly increases the membrane potential fluctuations, incontrast to the model with synaptic reversal potential. Absence of synaptic filtering
If synaptic filtering is neglected, h e , i become δ -functions: h e , i ( t ) = Ca e , i δ ( t ) , (40)where C is the membrane capacitance, and a e , i governsthe jump in the membrane potential ∆ V triggered by asingle pulse: ∆ V = ( E e , i − V )(1 − e − a e , i ) . (41)This model was studied extensively, e.g., in [23, 31, 32].In [23, 31], the formulas for the mean membrane potentialand its standard deviation are calculated in the diffusion FIG. 4.
The effect of membrane potential stabilization on spiking regularity in the Hodgkin-Huxley models.
A-C: Dependence of the C V of ISIs on the firing rate for different values of c (color-coded). The dotted parts of the curvesrepresent the sections where λ e >
100 kHz. In the subthreshold regimes, the output rate reaches its maximum and then startsdropping to zero. For the HH-DT model (C), the C V decreases at this point, whereas for the HH-0 (A) and HH-M (B) models,no clear improvement is observed. D-F: Contour plots with color-coded C V , c on the y -axis. If more than one input can producethe same PSFR with the same c , the lowest possible value of C V is color-coded. Note that there is no discontinuity in F, unlikein Fig. 3F. The transition to the more regular states with growing c is continuous, as is illustrated in Supplementary Figure 4.The data points were obtained from simulations with different input intensities λ e , λ i . approximation: (cid:104) V (cid:105) W = τ ( E L τ − L + E e λ e b e + E i λ i b i ) (42) σ V,W = τ L λ e b ( (cid:104) V (cid:105) − E e ) + λ i b ( (cid:104) V (cid:105) − E i ) τ L λ e b e (1 − b e /
2) + τ L λ i b i (1 − b i / , (43)where τ − = τ − L + λ e b e + λ i b i (44) b e , i = 1 − e − a e , i . (45)Richardson [23] reported that a higher inhibition-to-excitation ratio may lead to a decrease in the membranepotential fluctuations for strongly hyperpolarized mem-branes. However, the effect of inhibition reverses as themembrane potential depolarizes (Fig 6). Furthermore,the membrane potential does not stabilize within thelimit of infinite firing rates. Therefore, the time corre-lation of synaptic input introduced by synaptic filteringis necessary to observe the shunting effect of inhibitorysynapses. Regular firing in multicompartmental models
The models analyzed in this work are all single-compartmental models, i.e., models in which the chargeis distributed infinitely fast across the cell, and the mem-brane potential is therefore the same everywhere. In re-ality, neurons receive input predominantly at dendrites,and the spikes are initiated in the soma. To account forthis fact, multicompartmental models are typically em-ployed. The soma and dendritic parts can be modeled astwo separate compartments (for simplicity, as two identi-cal cylinders) connected through a coupling conductance g c : C d V S d t = − g L ( V S − E L ) − g c ( V S − V D ) (46) C d V D d t = − g L ( V D − E L ) − g c ( V D − V S ) −− a D ( g e ( V D − E e ) + g i ( V D − E i )) (47)where V S and V D are the membrane potentials of thesomatic and dendritic compartments, respectively; a D is the dendritic area; and V S is reset to V r when thethreshold θ is reached. FIG. 5.
Constant inhibition trajectories for the dy-namic threshold models.
In both the DT-LIF (A) andHH-DT (B) models, increasing the pre-synaptic inhibitory fir-ing rate (color) is beneficial for the firing regularity (measuredby C V , y -axis) for a wide range of PSFRs ( x -axis). In the hypothetical case of infinite input rates, V D = V ∞ ( c ) and V S periodically decay to V S = g L E L + g c V D g L + g c witha time constant τ = g L + g c a S C , resulting in regular ISIs T = − τ log (cid:18) V r − θV S − V r (cid:19) . (48)Therefore, it is possible to reach a wide range of firingrates with C V = 0 and decrease C V while maintaining aconstant mean firing rate by increasing c , similar to thecase of LIF with a dynamic threshold.The coupling conductance can be calculated as g c = d R a l [64], where d is the diameter of the cylinder, l isthe length, and R a = 150 Ω cm is the longitudinal resis-tance. If we consider that the original area of the neuronapproximately 3 . × − cm is split between the twocylinders and we set d = l , we obtain τ ≈ . µ s. Itis therefore unlikely that firing rate regularization withbiologically relevant post-synaptic firing rates would beobserved with biologically plausible inputs. CONCLUSION
We demonstrate that a higher inhibition-to-excitationratio and subsequently higher synaptic current fluctu- ations lead to a more stable membrane potential ifthe stimulation is modeled as time-filtered activationof synaptic conductances with reversal potentials. Ouranalysis thus provides a theoretical context for the exper-imental observations of [19]. Moreover, our results high-light the importance of incorporating synaptic filteringand reversal potentials into neuronal simulations. Thequalitative differences between neurons stimulated withwhite noise and colored noise current have been reportedin the literature [26–28]. However, we demonstrate thatrealistic synaptic filtering with reversal potentials is re-sponsible for a novel fluctuation-stabilization mechanismwhich cannot be observed in simplified models.We analyzed the impact of membrane potential sta-bilization on spike-firing regularity in GLIF models andHH-type models. We compared the effects of an increasedinhibition-to-excitation ratio on two different mecha-nisms of spike-firing adaptation: adaptation by a hyper-polarizing ionic current (AHP and M-current adaptation)and adaptation implemented as a dynamic firing thresh-old. Both SFA mechanisms are biologically relevant andare useful in neuronal modeling [39, 42, 47, 48, 65–68].We demonstrated that while an increase in inhibitionleads to less regular spike trains in the ionic current adap-tation models and models without any spike-firing adap-tation, it may enhance the firing regularity in the dy-namic threshold models. We observed this effect in boththe GLIF models and HH-type models.High presynaptic inhibitory activity is typical of corti-cal neurons. In the so called high-conductance state, to-tal inhibitory conductance can be several-fold larger thantotal excitatory conductance [11]. Our findings there-fore provide a novel view of the importance of the high-conductance state and inhibitory synapses in biologicalneural networks.
ACKNOWLEDGEMENTS
This work was supported by Charles University,project GA UK No. 1042120 and the Czech ScienceFoundation project 20-10251S.
Appendix A: Effective time constant approximation
The Eqs (1,2) can be rewritten as: Ca d V d t = − g ( V − E ) − g F e ( V − E e ) − g F i ( V − E i ) , (A1)where g = ag L + g + g , g , i , and g F e , i are the mean andfluctuating parts of the conductance input. The inputcan then be separated into its additive and multiplicativeparts: g F e ( V − E e ) = g F e ( E − E e ) + g F e ( V − E ) . (A2)By neglecting the multiplicative part g F e ( V − E ), weobtain the effective time constant approximation (ETA).0 FIG. 6.
Membrane potential with conductance input without synaptic filtering.
A: Membrane potential fluctuationsas a function of the mean membrane potential for different values of c = λ i b i λ e b e (color-coded), as calculated from the Eqs(42,43). The dashed line represents the limit lim λ e ,λ i →∞ σ V,W ( (cid:104) V (cid:105) W ). The membrane potential is not stabilized at infinite inputs.Above certain depolarizations, inhibition increases membrane potential fluctuations, contrary to the case of conductance inputwith synaptic filtering. B: Heatmap with color-coded standard deviation of the membrane potential. Parameters used were b e = 0 . b i = 0 . In the diffusion approximation, the mean and standarddeviation of the membrane potential are [23, 24]: (cid:104) V (cid:105) ETA = E , (A3) σ V, ETA = (cid:18) σ e g (cid:19) ( E e − E ) τ e τ e + τ ++ (cid:18) σ i g (cid:19) ( E i − E ) τ i τ i + τ , (A4)where τ = aCg is the effective time constant, and σ e , i arethe standard deviations of the excitatory and inhibitoryinputs. Appendix B: Limit cases of LIF and AHP-LIFmodelsHigh conductance limit of the LIF model If V ∞ ( c ) > θ , in the case of high input intensities, V ef is permanently above the threshold, and the effec-tive membrane time constant τ ( g e , g i ) approaches zero.Therefore, in the absence of a refractory period, the fir-ing rate f = µ ISI diverges ( µ ISI is the average ISI). If theaverage postsynaptic ISI is much shorter than synaptictimescales, we can assume that the input remains effec-tively constant during the entire ISI (corresponding tothe adiabatic approximation [28, 69–72]). The length ofthe ISI is then determined solely by the immediate valuesof the excitatory and inhibitory conductances T ( g e , g i ) = − aCag L + g e + g i log (cid:18) θ − V ef ( g e , g i ) V r − V ef ( g e , g i ) (cid:19) . (B1) Assuming independence of the inputs, the mean ISI andits standard deviation can then be approximated as µ ISI = − aCg tot log (cid:18) θ − E V r − E (cid:19) , (B2) σ = (cid:18) ∂T∂g e (cid:19) (cid:12)(cid:12)(cid:12) g e = g σ + (cid:18) ∂T∂g i (cid:19) (cid:12)(cid:12)(cid:12) g i = g σ = (B3)= ( aC ) g (cid:20) A e (( E e − E )( θ − V r ) + α ) g ( θ − E ) ( E − V r ) ++ cA i (( E i − E )( θ − V r ) + α ) g ( θ − E ) ( E − V r ) (cid:21) ,α = ( θ − E )( E − V r ) log E − θE − V r , where g tot = ag L + g + g (for validity of the approxima-tion see Fig. 7). Therefore, σ ISI /µ ISI = O (cid:16)(cid:0) g (cid:1) − / (cid:17) .We conclude that with growing input intensity, the firingrate diverges and C V → High conductance limit of the AHP-LIF model
Effective reversal potential
We follow the assumption that the fluctuations in V ef ( g e , g i ) are very small and therefore V AHPef (Eq. (7))is permanently above the threshold θ . With the ISI µ AHPISI (cid:28) τ AHP , g AHP ( t ) ≈ (cid:104) g AHP ( t ) (cid:105) = ∆ gτ AHP µ AHPISI . Analo-gously to the Eq. (B2)), we can use the following implicit1
FIG. 7.
Approximation of the PSFR and C V of LIF. Simulation results are color-coded. The dashed lines representthe approximations from the Eqs (B2,B3). The firing rate isapproximated very well (A). C V is approximated well for veryhigh PSFRs ( > t = 0 . <
100 Hz and ∆ t µ ISI
10 ms otherwise. equation to approximate the mean ISI: µ AHPISI = − aCg AHPtot log θ − E AHP0 V r − E AHP0 , , (B4)where g AHPtot = ag L + g + g + ∆ gτ AHP µ AHPISI , (B5) E AHP0 = ag L E L + g E e + g E i + ∆ gτ AHP µ AHPISI E K g AHPtot . (B6)We continue to evaluate the high-conductance limit of E AHP0 : lim g → + ∞ E AHP0 = V AHP ∞ ( c ) ≡≡ lim g → + ∞ ag L g E L + E e + cE i + τ AHP g µ AHPISI ∆ gE K ag L g + 1 + c + τ AHP g µ AHPISI ∆ g . (B7)Clearly, ag L g →
0. Therefore, it is important to evaluate the limit A = lim g → + ∞ g µ AHPISI . Then: V AHP ∞ ( c ) = E e + cE i + τ AHP A ∆ gE K c + τ AHP A ∆ g . (B8)By multiplying both sides of Eq. ((B4)) with g AHPtot andthen taking the limit of both sides of the equation, weobtain: ag L µ ISI + g (1 + c ) µ ISI + τ AHP ∆ g = − aC log θ − V AHP ∞ V r − V AHP ∞ , (B9) A (1 + c ) + τ AHP ∆ g = − aC log θ − V AHP ∞ V r − V AHP ∞ . (B10)Numerical solution of Eq. ((B10)) allows us to com-pare V AHP ∞ ( c ) with V ∞ ( c ) and thus provides a comparisonbetween the LIF model with and without the M-currentadaptation (Fig. 8). For approximately c > V ∞ ( c ) AHP ≈ θ . Therefore, the neuronrequires a very high input intensity for the fluctuations tobe so small that V AHPef permanently exceeds the thresh-old, and in the range of biologically feasible inputs, thefluctuations in V AHPef lead to bursting when V AHPef > θ and are silent when V AHPef ≤ θ (Fig. 8). The limit of C V Neglecting the variance of g AHP ( t ), the variance of ISIscan then be approximated analogously to Eq. ((B3)) as: σ = (cid:18) ∂µ ISI ∂g e (cid:19) σ + (cid:18) ∂µ ISI ∂g i (cid:19) σ , (B11)Our goal is to demonstrate that the coefficient of vari-ation (C V ) approaches zero. Using the definition of C V (Eq. (26)) and Eq. (B4), we havelim g → + ∞ C V = lim g → + ∞ σ ISI µ ISI (B12)= − lim g → + ∞ g AHPtot aC σ
ISI log θ − E AHP0 V r − E AHP0 (B13)= − lim g → + ∞ g AHPtot σ ISI lim g → + ∞ (cid:18) log θ − E AHP0 V r − E AHP0 (cid:19) − . (B14)Since −∞ < lim g → + ∞ (cid:16) log θ − E AHP0 V r − E AHP0 (cid:17) − <
0, it remains tobe shown that lim g → + ∞ g AHPtot σ ISI = 0, which can be shownby using the implicit differentiation formula.2
FIG. 8.
High conductance limit of the AHP-LIF model
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T h e o r a n g e li n e s r e p r e s e n tt h e m o d i fi e d r a t e f un c t i o n s f o r t h e HH - D T m o d e l. A s a n e ff ec t o f t h i s m o d i fi c a t i o n , t h e r ec o v e r y t i m e i n t h e HH - D T m o d e li s p r o l o n g e d ( C ) , w h il e t h e s t e a d y s t a t e a c t i v a t i o n h ∞ r e m a i n s r e l a t i v e l y un c h a n g e d ( D ) . E : C o m pu t a t i o n o f t h e t h r e s h o l d . I f n oa dd i t i o n a l p r e - s y n a p t i c s p i k e s a rr i v e a f t e r t = . m s , n o s p i k e i s o b s e r v e d ( g r ee n t r a ce ) . I f t h e p r e - s y n a p t i c s p i k e ss t o p a rr i v i n g ∆ t = . m s l a t e r , a s p i k e i s o b s e r v e d . T h e r e f o r e , w e d e fin e t h e t h r e s h o l d a s b e i n g r e a c h e d a t t = . m s . F - H : C o m p a r i s o n o f t h ec a l c u l a t e d t h r e s h o l d w i t h t h e p r ece d i n g i n t e r s p i k e i n t e r v a l ( I S I ) . W e v a li d a t e d t h a tt h e t h r e s h o l dd ec r e a s e s w i t h t i m e a f t e r a s p i k e i s fi r e db y e s t i m a t i n g t h e fi r i n g t h r e s h o l d i nd i v i du a ll y f o r e a c h s p i k e du r i n ga s i m u l a t i o n w i t h a s t a t i o n a r y i npu t , r e s u l t i n g i n a p o s t - s y n a p t i c fi r i n g r a t e o f a pp r o x i m a t e l y H z N e i t h e r t h e n o n - a d a p t i n g m o d e l ( HH - , p a n e l F ) n o r t h e m o d e l w i t h M - c u rr e n t a d a p t a t i o n ( HH - M , p a n e l G ) d i s p l a y ss i g n s o f s i g n i fi c a n t d e p e nd e n ce b e t w ee n t h e t h r e s h o l d a nd I S I . F o r t h e d y n a m i c t h r e s h o l d m o d e l ( HH - D T , p a n e l H ) w e o b s e r v e d a c l e a r d ec r e a s e i n t h e t h r e s h o l d w i t h i n c r e a s i n g du r a t i o n o f t h e p r ece d i n g I S I . T h e r e d li n e , a c t i n ga s a v i s u a l a i d , w a s o b t a i n e d b y a K NN r e g r e ss i o n . upp l e m e n t a r y F i g u r e : V a r y i n g t h e m e m b r a n e t i m e c o n s t a n t . T o p r o w : T h e m e m b r a n e t i m ec o n s t a n t i s v a r i e db y c h a n g i n g t h e l e a k c o ndu c t a n ce g L . W i t h g r e a t e r l e a k c o ndu c t a n ce , t h e m e m b r a n e p o t e n t i a l V f o ll o w s V e f m o r ec l o s e l y a nd t h e nh i g h e r c l e a d s t o l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s m o r e un i v e r s a ll y ( a pp r o x i m a t i o n o f σ V , c a l c u l a t e d f r o m E T A ,i s c o l o r c o d e d ) . T h e r e g i o n s w h e r e h i g h e r c d ec r e a s e s / i n c r e a s e s t h e m e m b r a n e p o t e n t i a l a r e s e p a r a t e db y t h e d o tt e d li n e . B o tt o m r o w : T h e s t r u c t u r e o f t h ec o n t o u r p l o t s i nd i c a t i n g h o w fi r i n g r e g u l a r i t y ( C V , c o l o r c o d e d ) c h a n g e s w i t h c a nd P S F R f o ll o w s t h e s t r u c t u r e o f t h e m e m b r a n e p o t e n t i a l flu c t u a t i o nh e a t m a p s (t o p r o w ) . W i t hh i g h e r g L , h i g h e r c i n c r e a s e s t h e fi r i n g r e g u l a r i t y f o r a w i d e rr a n g e o f fi r i n g r a t e s a nd a w i d e rr a n g e o f c . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r fi r i n g r e g u l a r i t y . upp l e m e n t a r y F i g u r e : V a r y i n g t h e a p li t ud e o f i nh i b i t o r y s y n a p s e s . W i t h g r e a t e r a m p li t ud e o f t h e i nh i b i t o r y s y n a p s e s ( A i , E q . ( )) , t h e s t a b ili s a t i o n o f t h e m e m b r a n e p o t e n t i a li s o b s e r v e d o n l y f o r h i g h e r v a l u e s o f c (t o p r o w ) . T h i s i s r e fl ec t e d i n t h e fi r i n g r e g u l a r i t y o f t h e D T - L I F m o d e l ( b o tt o m r o w ) , a n a l ogo u s l y t o t h e Supp l e m e n t a r y F i g u r e . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s ( f o r t h e t o p r o w ) o r h i g h e r / l o w e r fi r i n g r e g u l a r i t y ( f o r t h e b o tt o m r o w ) . upp l e m e n t a r y F i g u r e : C o n t i nu o u s c h a n g e o f C V i n t h e HH - D T m o d e l. L e f t : S a m e a s F i g . C , bu t f o r c ∈ { , . , . , . , . } . D a s h e d a ndd o tt e d v e r t i c a lli n e s w i t hp o i n t s i nd i c a t ee
Neuronal Dy-namics (Cambridge University Press, 2019).[47] C. Teeter, R. Iyer, V. Menon, N. Gouwens, D. Feng,J. Berg, A. Szafer, N. Cain, H. Zeng, M. Hawrylycz,C. Koch, and S. Mihalas, Generalized leaky integrate-and-fire models classify multiple neuron types, Nat. Com-mun , 709 (2018).[48] J. Benda, L. Maler, and A. Longtin, Linear Versus Non-linear Signal Transmission in Neuron Models With Adap-tation Currents or Dynamic Thresholds, J. Neurophysiol , 2806 (2010).[49] A. Destexhe and D. Par´e, Impact of Network Activityon the Integrative Properties of Neocortical PyramidalNeurons In Vivo, J. Neurophysiol , 1531 (1999).[50] W. R. Softky and C. Koch, The highly irregular firing ofcortical cells is inconsistent with temporal integration ofrandom EPSPs, J. Neurosci , 334 (1993).[51] D. P. Heyman and M. J. Sobel, Stochastic Models inOperations Research: Stochastic Processes and Operat- ing Characteristics , Dover Books on Computer ScienceSeries (Dover Publications, 2004).[52] B. Lindner, Superposition of many independent spiketrains is generally not a Poisson process, Phys Rev E , 022901 (2006).[53] J. F. C. Kingman, Poisson Processes (Oxford Studies inProbability) (Clarendon Press, 1993).[54] D. J. Amit and M. V. Tsodyks, Effective neurons and at-tractor neural networks in cortical environment, Network , 121 (1992).[55] T. Toyoizumi, K. Aihara, and S.-i. Amari, Fisher Infor-mation for Spike-Based Population Decoding, Phys. Rev.Lett. , 098102 (2006).[56] L. Kostal, P. Lansky, and J.-P. Rospars, Neuronal cod-ing and spiking randomness, Eur. J. Neurosci , 2693(2007).[57] R. R. de Ruyter van Steveninck, Reproducibility andVariability in Neural Spike Trains, Science , 1805(1997).[58] S. P. Strong, R. Koberle, R. R. de Ruyter van Steveninck,and W. Bialek, Entropy and Information in Neural SpikeTrains, Phys. Rev. Lett. , 197 (1998).[59] P. H. E. Tiesinga, J. V. Jos´e, and T. J. Sejnowski, Com-parison of current-driven and conductance-driven neocor-tical model neurons with Hodgkin-Huxley voltage-gatedchannels, Phys Rev E , 8413 (2000).[60] M. J. Chacron, B. Lindner, and A. Longtin, Noise Shap-ing by Interval Correlations Increases Information Trans-fer, Phys. Rev. Lett. , 080601 (2004).[61] B. Lindner, M. J. Chacron, and A. Longtin, Integrate-and-fire neurons with threshold noise: A tractable modelof how interspike interval correlations affect neuronal sig-nal transmission, Phys Rev E , 021911 (2005).[62] F. Farkhooi, E. Muller, and M. P. Nawrot, Adaptationreduces variability of the neuronal population code, PhysRev E , 050905 (2011).[63] B. Lindner and A. Longtin, Comment on “Characteri-zation of Subthreshold Voltage Fluctuations in NeuronalMembranes,” by M. Rudolph and A. Destexhe, NeuralComput. , 1896 (2006).[64] D. Sterratt, B. Graham, A. Gillies, and D. Willshaw, Principles of Computational Modelling in Neuroscience (Cambridge University Press, 2011).[65] M. J. Chacron, A. Longtin, M. St-Hilaire, and L. Maler,Suprathreshold Stochastic Firing Dynamics with Mem-ory in P-Type Electroreceptors, Phys. Rev. Lett. , 1576(2000).[66] R. Kobayashi, Y. Tsubo, and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci. , 9 (2009).[67] W. Gerstner and R. Naud, How Good Are Neuron Mod-els?, Science , 379 (2009).[68] R. Kobayashi, S. Kurita, A. Kurth, K. Kitano,K. Mizuseki, M. Diesmann, B. J. Richmond, and S. Shi-nomoto, Reconstructing neuronal circuitry from parallelspike trains, Nat. Commun. , 4468 (2019).[69] R. Moreno-Bote and N. Parga, Membrane Potential andResponse Properties of Populations of Cortical Neuronsin the High Conductance State, Phys. Rev. Lett. ,088103 (2005).[70] R. Moreno-Bote and N. Parga, Auto- and Crosscorrel-ograms for the Spike Response of Leaky Integrate-and-Fire Neurons with Slow Synapses, Phys. Rev. Lett. , 028101 (2006).[71] R. Moreno-Bote, A. Renart, and N. Parga, Theory of In-put Spike Auto- and Cross-Correlations and Their Effecton the Response of Spiking Neurons, Neural Comput. ,1651 (2008).[72] R. Moreno-Bote and N. Parga, Response of Integrate-and-Fire Neurons to Noisy Inputs Filtered by Synapseswith Arbitrary Timescales: Firing Rate and Correlations,Neural Comput. , 1528 (2010). upp l e m e n t a r y F i g u r e : C o n s t r u c t i o n a nd v a li d a t i o n o f d y n a m i c t h r e s h o l d HH m o d e l. A - D : T h e b l u e li n e s r e p r e s e n tt h e a c t i v a t i o n ( A ) a nd i n a c t i v a t i o n ( B ) r a t e f un c t i o n s o f t h e ga t i n g v a r i a b l e h o f t h e n o n - a d a p t i n g HH m o d e l ( HH - ) . T h e o r a n g e li n e s r e p r e s e n tt h e m o d i fi e d r a t e f un c t i o n s f o r t h e HH - D T m o d e l. A s a n e ff ec t o f t h i s m o d i fi c a t i o n , t h e r ec o v e r y t i m e i n t h e HH - D T m o d e li s p r o l o n g e d ( C ) , w h il e t h e s t e a d y s t a t e a c t i v a t i o n h ∞ r e m a i n s r e l a t i v e l y un c h a n g e d ( D ) . E : C o m pu t a t i o n o f t h e t h r e s h o l d . I f n oa dd i t i o n a l p r e - s y n a p t i c s p i k e s a rr i v e a f t e r t = . m s , n o s p i k e i s o b s e r v e d ( g r ee n t r a ce ) . I f t h e p r e - s y n a p t i c s p i k e ss t o p a rr i v i n g ∆ t = . m s l a t e r , a s p i k e i s o b s e r v e d . T h e r e f o r e , w e d e fin e t h e t h r e s h o l d a s b e i n g r e a c h e d a t t = . m s . F - H : C o m p a r i s o n o f t h ec a l c u l a t e d t h r e s h o l d w i t h t h e p r ece d i n g i n t e r s p i k e i n t e r v a l ( I S I ) . W e v a li d a t e d t h a tt h e t h r e s h o l dd ec r e a s e s w i t h t i m e a f t e r a s p i k e i s fi r e db y e s t i m a t i n g t h e fi r i n g t h r e s h o l d i nd i v i du a ll y f o r e a c h s p i k e du r i n ga s i m u l a t i o n w i t h a s t a t i o n a r y i npu t , r e s u l t i n g i n a p o s t - s y n a p t i c fi r i n g r a t e o f a pp r o x i m a t e l y H z N e i t h e r t h e n o n - a d a p t i n g m o d e l ( HH - , p a n e l F ) n o r t h e m o d e l w i t h M - c u rr e n t a d a p t a t i o n ( HH - M , p a n e l G ) d i s p l a y ss i g n s o f s i g n i fi c a n t d e p e nd e n ce b e t w ee n t h e t h r e s h o l d a nd I S I . F o r t h e d y n a m i c t h r e s h o l d m o d e l ( HH - D T , p a n e l H ) w e o b s e r v e d a c l e a r d ec r e a s e i n t h e t h r e s h o l d w i t h i n c r e a s i n g du r a t i o n o f t h e p r ece d i n g I S I . T h e r e d li n e , a c t i n ga s a v i s u a l a i d , w a s o b t a i n e d b y a K NN r e g r e ss i o n . upp l e m e n t a r y F i g u r e : V a r y i n g t h e m e m b r a n e t i m e c o n s t a n t . T o p r o w : T h e m e m b r a n e t i m ec o n s t a n t i s v a r i e db y c h a n g i n g t h e l e a k c o ndu c t a n ce g L . W i t h g r e a t e r l e a k c o ndu c t a n ce , t h e m e m b r a n e p o t e n t i a l V f o ll o w s V e f m o r ec l o s e l y a nd t h e nh i g h e r c l e a d s t o l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s m o r e un i v e r s a ll y ( a pp r o x i m a t i o n o f σ V , c a l c u l a t e d f r o m E T A ,i s c o l o r c o d e d ) . T h e r e g i o n s w h e r e h i g h e r c d ec r e a s e s / i n c r e a s e s t h e m e m b r a n e p o t e n t i a l a r e s e p a r a t e db y t h e d o tt e d li n e . B o tt o m r o w : T h e s t r u c t u r e o f t h ec o n t o u r p l o t s i nd i c a t i n g h o w fi r i n g r e g u l a r i t y ( C V , c o l o r c o d e d ) c h a n g e s w i t h c a nd P S F R f o ll o w s t h e s t r u c t u r e o f t h e m e m b r a n e p o t e n t i a l flu c t u a t i o nh e a t m a p s (t o p r o w ) . W i t hh i g h e r g L , h i g h e r c i n c r e a s e s t h e fi r i n g r e g u l a r i t y f o r a w i d e rr a n g e o f fi r i n g r a t e s a nd a w i d e rr a n g e o f c . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r fi r i n g r e g u l a r i t y . upp l e m e n t a r y F i g u r e : V a r y i n g t h e a p li t ud e o f i nh i b i t o r y s y n a p s e s . W i t h g r e a t e r a m p li t ud e o f t h e i nh i b i t o r y s y n a p s e s ( A i , E q . ( )) , t h e s t a b ili s a t i o n o f t h e m e m b r a n e p o t e n t i a li s o b s e r v e d o n l y f o r h i g h e r v a l u e s o f c (t o p r o w ) . T h i s i s r e fl ec t e d i n t h e fi r i n g r e g u l a r i t y o f t h e D T - L I F m o d e l ( b o tt o m r o w ) , a n a l ogo u s l y t o t h e Supp l e m e n t a r y F i g u r e . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s ( f o r t h e t o p r o w ) o r h i g h e r / l o w e r fi r i n g r e g u l a r i t y ( f o r t h e b o tt o m r o w ) . upp l e m e n t a r y F i g u r e : C o n t i nu o u s c h a n g e o f C V i n t h e HH - D T m o d e l. L e f t : S a m e a s F i g . C , bu t f o r c ∈ { , . , . , . , . } . D a s h e d a ndd o tt e d v e r t i c a lli n e s w i t hp o i n t s i nd i c a t ee t h ee
Neuronal Dy-namics (Cambridge University Press, 2019).[47] C. Teeter, R. Iyer, V. Menon, N. Gouwens, D. Feng,J. Berg, A. Szafer, N. Cain, H. Zeng, M. Hawrylycz,C. Koch, and S. Mihalas, Generalized leaky integrate-and-fire models classify multiple neuron types, Nat. Com-mun , 709 (2018).[48] J. Benda, L. Maler, and A. Longtin, Linear Versus Non-linear Signal Transmission in Neuron Models With Adap-tation Currents or Dynamic Thresholds, J. Neurophysiol , 2806 (2010).[49] A. Destexhe and D. Par´e, Impact of Network Activityon the Integrative Properties of Neocortical PyramidalNeurons In Vivo, J. Neurophysiol , 1531 (1999).[50] W. R. Softky and C. Koch, The highly irregular firing ofcortical cells is inconsistent with temporal integration ofrandom EPSPs, J. Neurosci , 334 (1993).[51] D. P. Heyman and M. J. Sobel, Stochastic Models inOperations Research: Stochastic Processes and Operat- ing Characteristics , Dover Books on Computer ScienceSeries (Dover Publications, 2004).[52] B. Lindner, Superposition of many independent spiketrains is generally not a Poisson process, Phys Rev E , 022901 (2006).[53] J. F. C. Kingman, Poisson Processes (Oxford Studies inProbability) (Clarendon Press, 1993).[54] D. J. Amit and M. V. Tsodyks, Effective neurons and at-tractor neural networks in cortical environment, Network , 121 (1992).[55] T. Toyoizumi, K. Aihara, and S.-i. Amari, Fisher Infor-mation for Spike-Based Population Decoding, Phys. Rev.Lett. , 098102 (2006).[56] L. Kostal, P. Lansky, and J.-P. Rospars, Neuronal cod-ing and spiking randomness, Eur. J. Neurosci , 2693(2007).[57] R. R. de Ruyter van Steveninck, Reproducibility andVariability in Neural Spike Trains, Science , 1805(1997).[58] S. P. Strong, R. Koberle, R. R. de Ruyter van Steveninck,and W. Bialek, Entropy and Information in Neural SpikeTrains, Phys. Rev. Lett. , 197 (1998).[59] P. H. E. Tiesinga, J. V. Jos´e, and T. J. Sejnowski, Com-parison of current-driven and conductance-driven neocor-tical model neurons with Hodgkin-Huxley voltage-gatedchannels, Phys Rev E , 8413 (2000).[60] M. J. Chacron, B. Lindner, and A. Longtin, Noise Shap-ing by Interval Correlations Increases Information Trans-fer, Phys. Rev. Lett. , 080601 (2004).[61] B. Lindner, M. J. Chacron, and A. Longtin, Integrate-and-fire neurons with threshold noise: A tractable modelof how interspike interval correlations affect neuronal sig-nal transmission, Phys Rev E , 021911 (2005).[62] F. Farkhooi, E. Muller, and M. P. Nawrot, Adaptationreduces variability of the neuronal population code, PhysRev E , 050905 (2011).[63] B. Lindner and A. Longtin, Comment on “Characteri-zation of Subthreshold Voltage Fluctuations in NeuronalMembranes,” by M. Rudolph and A. Destexhe, NeuralComput. , 1896 (2006).[64] D. Sterratt, B. Graham, A. Gillies, and D. Willshaw, Principles of Computational Modelling in Neuroscience (Cambridge University Press, 2011).[65] M. J. Chacron, A. Longtin, M. St-Hilaire, and L. Maler,Suprathreshold Stochastic Firing Dynamics with Mem-ory in P-Type Electroreceptors, Phys. Rev. Lett. , 1576(2000).[66] R. Kobayashi, Y. Tsubo, and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci. , 9 (2009).[67] W. Gerstner and R. Naud, How Good Are Neuron Mod-els?, Science , 379 (2009).[68] R. Kobayashi, S. Kurita, A. Kurth, K. Kitano,K. Mizuseki, M. Diesmann, B. J. Richmond, and S. Shi-nomoto, Reconstructing neuronal circuitry from parallelspike trains, Nat. Commun. , 4468 (2019).[69] R. Moreno-Bote and N. Parga, Membrane Potential andResponse Properties of Populations of Cortical Neuronsin the High Conductance State, Phys. Rev. Lett. ,088103 (2005).[70] R. Moreno-Bote and N. Parga, Auto- and Crosscorrel-ograms for the Spike Response of Leaky Integrate-and-Fire Neurons with Slow Synapses, Phys. Rev. Lett. , 028101 (2006).[71] R. Moreno-Bote, A. Renart, and N. Parga, Theory of In-put Spike Auto- and Cross-Correlations and Their Effecton the Response of Spiking Neurons, Neural Comput. ,1651 (2008).[72] R. Moreno-Bote and N. Parga, Response of Integrate-and-Fire Neurons to Noisy Inputs Filtered by Synapseswith Arbitrary Timescales: Firing Rate and Correlations,Neural Comput. , 1528 (2010). upp l e m e n t a r y F i g u r e : C o n s t r u c t i o n a nd v a li d a t i o n o f d y n a m i c t h r e s h o l d HH m o d e l. A - D : T h e b l u e li n e s r e p r e s e n tt h e a c t i v a t i o n ( A ) a nd i n a c t i v a t i o n ( B ) r a t e f un c t i o n s o f t h e ga t i n g v a r i a b l e h o f t h e n o n - a d a p t i n g HH m o d e l ( HH - ) . T h e o r a n g e li n e s r e p r e s e n tt h e m o d i fi e d r a t e f un c t i o n s f o r t h e HH - D T m o d e l. A s a n e ff ec t o f t h i s m o d i fi c a t i o n , t h e r ec o v e r y t i m e i n t h e HH - D T m o d e li s p r o l o n g e d ( C ) , w h il e t h e s t e a d y s t a t e a c t i v a t i o n h ∞ r e m a i n s r e l a t i v e l y un c h a n g e d ( D ) . E : C o m pu t a t i o n o f t h e t h r e s h o l d . I f n oa dd i t i o n a l p r e - s y n a p t i c s p i k e s a rr i v e a f t e r t = . m s , n o s p i k e i s o b s e r v e d ( g r ee n t r a ce ) . I f t h e p r e - s y n a p t i c s p i k e ss t o p a rr i v i n g ∆ t = . m s l a t e r , a s p i k e i s o b s e r v e d . T h e r e f o r e , w e d e fin e t h e t h r e s h o l d a s b e i n g r e a c h e d a t t = . m s . F - H : C o m p a r i s o n o f t h ec a l c u l a t e d t h r e s h o l d w i t h t h e p r ece d i n g i n t e r s p i k e i n t e r v a l ( I S I ) . W e v a li d a t e d t h a tt h e t h r e s h o l dd ec r e a s e s w i t h t i m e a f t e r a s p i k e i s fi r e db y e s t i m a t i n g t h e fi r i n g t h r e s h o l d i nd i v i du a ll y f o r e a c h s p i k e du r i n ga s i m u l a t i o n w i t h a s t a t i o n a r y i npu t , r e s u l t i n g i n a p o s t - s y n a p t i c fi r i n g r a t e o f a pp r o x i m a t e l y H z N e i t h e r t h e n o n - a d a p t i n g m o d e l ( HH - , p a n e l F ) n o r t h e m o d e l w i t h M - c u rr e n t a d a p t a t i o n ( HH - M , p a n e l G ) d i s p l a y ss i g n s o f s i g n i fi c a n t d e p e nd e n ce b e t w ee n t h e t h r e s h o l d a nd I S I . F o r t h e d y n a m i c t h r e s h o l d m o d e l ( HH - D T , p a n e l H ) w e o b s e r v e d a c l e a r d ec r e a s e i n t h e t h r e s h o l d w i t h i n c r e a s i n g du r a t i o n o f t h e p r ece d i n g I S I . T h e r e d li n e , a c t i n ga s a v i s u a l a i d , w a s o b t a i n e d b y a K NN r e g r e ss i o n . upp l e m e n t a r y F i g u r e : V a r y i n g t h e m e m b r a n e t i m e c o n s t a n t . T o p r o w : T h e m e m b r a n e t i m ec o n s t a n t i s v a r i e db y c h a n g i n g t h e l e a k c o ndu c t a n ce g L . W i t h g r e a t e r l e a k c o ndu c t a n ce , t h e m e m b r a n e p o t e n t i a l V f o ll o w s V e f m o r ec l o s e l y a nd t h e nh i g h e r c l e a d s t o l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s m o r e un i v e r s a ll y ( a pp r o x i m a t i o n o f σ V , c a l c u l a t e d f r o m E T A ,i s c o l o r c o d e d ) . T h e r e g i o n s w h e r e h i g h e r c d ec r e a s e s / i n c r e a s e s t h e m e m b r a n e p o t e n t i a l a r e s e p a r a t e db y t h e d o tt e d li n e . B o tt o m r o w : T h e s t r u c t u r e o f t h ec o n t o u r p l o t s i nd i c a t i n g h o w fi r i n g r e g u l a r i t y ( C V , c o l o r c o d e d ) c h a n g e s w i t h c a nd P S F R f o ll o w s t h e s t r u c t u r e o f t h e m e m b r a n e p o t e n t i a l flu c t u a t i o nh e a t m a p s (t o p r o w ) . W i t hh i g h e r g L , h i g h e r c i n c r e a s e s t h e fi r i n g r e g u l a r i t y f o r a w i d e rr a n g e o f fi r i n g r a t e s a nd a w i d e rr a n g e o f c . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r fi r i n g r e g u l a r i t y . upp l e m e n t a r y F i g u r e : V a r y i n g t h e a p li t ud e o f i nh i b i t o r y s y n a p s e s . W i t h g r e a t e r a m p li t ud e o f t h e i nh i b i t o r y s y n a p s e s ( A i , E q . ( )) , t h e s t a b ili s a t i o n o f t h e m e m b r a n e p o t e n t i a li s o b s e r v e d o n l y f o r h i g h e r v a l u e s o f c (t o p r o w ) . T h i s i s r e fl ec t e d i n t h e fi r i n g r e g u l a r i t y o f t h e D T - L I F m o d e l ( b o tt o m r o w ) , a n a l ogo u s l y t o t h e Supp l e m e n t a r y F i g u r e . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s ( f o r t h e t o p r o w ) o r h i g h e r / l o w e r fi r i n g r e g u l a r i t y ( f o r t h e b o tt o m r o w ) . upp l e m e n t a r y F i g u r e : C o n t i nu o u s c h a n g e o f C V i n t h e HH - D T m o d e l. L e f t : S a m e a s F i g . C , bu t f o r c ∈ { , . , . , . , . } . D a s h e d a ndd o tt e d v e r t i c a lli n e s w i t hp o i n t s i nd i c a t ee t h ee l o w ee
Neuronal Dy-namics (Cambridge University Press, 2019).[47] C. Teeter, R. Iyer, V. Menon, N. Gouwens, D. Feng,J. Berg, A. Szafer, N. Cain, H. Zeng, M. Hawrylycz,C. Koch, and S. Mihalas, Generalized leaky integrate-and-fire models classify multiple neuron types, Nat. Com-mun , 709 (2018).[48] J. Benda, L. Maler, and A. Longtin, Linear Versus Non-linear Signal Transmission in Neuron Models With Adap-tation Currents or Dynamic Thresholds, J. Neurophysiol , 2806 (2010).[49] A. Destexhe and D. Par´e, Impact of Network Activityon the Integrative Properties of Neocortical PyramidalNeurons In Vivo, J. Neurophysiol , 1531 (1999).[50] W. R. Softky and C. Koch, The highly irregular firing ofcortical cells is inconsistent with temporal integration ofrandom EPSPs, J. Neurosci , 334 (1993).[51] D. P. Heyman and M. J. Sobel, Stochastic Models inOperations Research: Stochastic Processes and Operat- ing Characteristics , Dover Books on Computer ScienceSeries (Dover Publications, 2004).[52] B. Lindner, Superposition of many independent spiketrains is generally not a Poisson process, Phys Rev E , 022901 (2006).[53] J. F. C. Kingman, Poisson Processes (Oxford Studies inProbability) (Clarendon Press, 1993).[54] D. J. Amit and M. V. Tsodyks, Effective neurons and at-tractor neural networks in cortical environment, Network , 121 (1992).[55] T. Toyoizumi, K. Aihara, and S.-i. Amari, Fisher Infor-mation for Spike-Based Population Decoding, Phys. Rev.Lett. , 098102 (2006).[56] L. Kostal, P. Lansky, and J.-P. Rospars, Neuronal cod-ing and spiking randomness, Eur. J. Neurosci , 2693(2007).[57] R. R. de Ruyter van Steveninck, Reproducibility andVariability in Neural Spike Trains, Science , 1805(1997).[58] S. P. Strong, R. Koberle, R. R. de Ruyter van Steveninck,and W. Bialek, Entropy and Information in Neural SpikeTrains, Phys. Rev. Lett. , 197 (1998).[59] P. H. E. Tiesinga, J. V. Jos´e, and T. J. Sejnowski, Com-parison of current-driven and conductance-driven neocor-tical model neurons with Hodgkin-Huxley voltage-gatedchannels, Phys Rev E , 8413 (2000).[60] M. J. Chacron, B. Lindner, and A. Longtin, Noise Shap-ing by Interval Correlations Increases Information Trans-fer, Phys. Rev. Lett. , 080601 (2004).[61] B. Lindner, M. J. Chacron, and A. Longtin, Integrate-and-fire neurons with threshold noise: A tractable modelof how interspike interval correlations affect neuronal sig-nal transmission, Phys Rev E , 021911 (2005).[62] F. Farkhooi, E. Muller, and M. P. Nawrot, Adaptationreduces variability of the neuronal population code, PhysRev E , 050905 (2011).[63] B. Lindner and A. Longtin, Comment on “Characteri-zation of Subthreshold Voltage Fluctuations in NeuronalMembranes,” by M. Rudolph and A. Destexhe, NeuralComput. , 1896 (2006).[64] D. Sterratt, B. Graham, A. Gillies, and D. Willshaw, Principles of Computational Modelling in Neuroscience (Cambridge University Press, 2011).[65] M. J. Chacron, A. Longtin, M. St-Hilaire, and L. Maler,Suprathreshold Stochastic Firing Dynamics with Mem-ory in P-Type Electroreceptors, Phys. Rev. Lett. , 1576(2000).[66] R. Kobayashi, Y. Tsubo, and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci. , 9 (2009).[67] W. Gerstner and R. Naud, How Good Are Neuron Mod-els?, Science , 379 (2009).[68] R. Kobayashi, S. Kurita, A. Kurth, K. Kitano,K. Mizuseki, M. Diesmann, B. J. Richmond, and S. Shi-nomoto, Reconstructing neuronal circuitry from parallelspike trains, Nat. Commun. , 4468 (2019).[69] R. Moreno-Bote and N. Parga, Membrane Potential andResponse Properties of Populations of Cortical Neuronsin the High Conductance State, Phys. Rev. Lett. ,088103 (2005).[70] R. Moreno-Bote and N. Parga, Auto- and Crosscorrel-ograms for the Spike Response of Leaky Integrate-and-Fire Neurons with Slow Synapses, Phys. Rev. Lett. , 028101 (2006).[71] R. Moreno-Bote, A. Renart, and N. Parga, Theory of In-put Spike Auto- and Cross-Correlations and Their Effecton the Response of Spiking Neurons, Neural Comput. ,1651 (2008).[72] R. Moreno-Bote and N. Parga, Response of Integrate-and-Fire Neurons to Noisy Inputs Filtered by Synapseswith Arbitrary Timescales: Firing Rate and Correlations,Neural Comput. , 1528 (2010). upp l e m e n t a r y F i g u r e : C o n s t r u c t i o n a nd v a li d a t i o n o f d y n a m i c t h r e s h o l d HH m o d e l. A - D : T h e b l u e li n e s r e p r e s e n tt h e a c t i v a t i o n ( A ) a nd i n a c t i v a t i o n ( B ) r a t e f un c t i o n s o f t h e ga t i n g v a r i a b l e h o f t h e n o n - a d a p t i n g HH m o d e l ( HH - ) . T h e o r a n g e li n e s r e p r e s e n tt h e m o d i fi e d r a t e f un c t i o n s f o r t h e HH - D T m o d e l. A s a n e ff ec t o f t h i s m o d i fi c a t i o n , t h e r ec o v e r y t i m e i n t h e HH - D T m o d e li s p r o l o n g e d ( C ) , w h il e t h e s t e a d y s t a t e a c t i v a t i o n h ∞ r e m a i n s r e l a t i v e l y un c h a n g e d ( D ) . E : C o m pu t a t i o n o f t h e t h r e s h o l d . I f n oa dd i t i o n a l p r e - s y n a p t i c s p i k e s a rr i v e a f t e r t = . m s , n o s p i k e i s o b s e r v e d ( g r ee n t r a ce ) . I f t h e p r e - s y n a p t i c s p i k e ss t o p a rr i v i n g ∆ t = . m s l a t e r , a s p i k e i s o b s e r v e d . T h e r e f o r e , w e d e fin e t h e t h r e s h o l d a s b e i n g r e a c h e d a t t = . m s . F - H : C o m p a r i s o n o f t h ec a l c u l a t e d t h r e s h o l d w i t h t h e p r ece d i n g i n t e r s p i k e i n t e r v a l ( I S I ) . W e v a li d a t e d t h a tt h e t h r e s h o l dd ec r e a s e s w i t h t i m e a f t e r a s p i k e i s fi r e db y e s t i m a t i n g t h e fi r i n g t h r e s h o l d i nd i v i du a ll y f o r e a c h s p i k e du r i n ga s i m u l a t i o n w i t h a s t a t i o n a r y i npu t , r e s u l t i n g i n a p o s t - s y n a p t i c fi r i n g r a t e o f a pp r o x i m a t e l y H z N e i t h e r t h e n o n - a d a p t i n g m o d e l ( HH - , p a n e l F ) n o r t h e m o d e l w i t h M - c u rr e n t a d a p t a t i o n ( HH - M , p a n e l G ) d i s p l a y ss i g n s o f s i g n i fi c a n t d e p e nd e n ce b e t w ee n t h e t h r e s h o l d a nd I S I . F o r t h e d y n a m i c t h r e s h o l d m o d e l ( HH - D T , p a n e l H ) w e o b s e r v e d a c l e a r d ec r e a s e i n t h e t h r e s h o l d w i t h i n c r e a s i n g du r a t i o n o f t h e p r ece d i n g I S I . T h e r e d li n e , a c t i n ga s a v i s u a l a i d , w a s o b t a i n e d b y a K NN r e g r e ss i o n . upp l e m e n t a r y F i g u r e : V a r y i n g t h e m e m b r a n e t i m e c o n s t a n t . T o p r o w : T h e m e m b r a n e t i m ec o n s t a n t i s v a r i e db y c h a n g i n g t h e l e a k c o ndu c t a n ce g L . W i t h g r e a t e r l e a k c o ndu c t a n ce , t h e m e m b r a n e p o t e n t i a l V f o ll o w s V e f m o r ec l o s e l y a nd t h e nh i g h e r c l e a d s t o l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s m o r e un i v e r s a ll y ( a pp r o x i m a t i o n o f σ V , c a l c u l a t e d f r o m E T A ,i s c o l o r c o d e d ) . T h e r e g i o n s w h e r e h i g h e r c d ec r e a s e s / i n c r e a s e s t h e m e m b r a n e p o t e n t i a l a r e s e p a r a t e db y t h e d o tt e d li n e . B o tt o m r o w : T h e s t r u c t u r e o f t h ec o n t o u r p l o t s i nd i c a t i n g h o w fi r i n g r e g u l a r i t y ( C V , c o l o r c o d e d ) c h a n g e s w i t h c a nd P S F R f o ll o w s t h e s t r u c t u r e o f t h e m e m b r a n e p o t e n t i a l flu c t u a t i o nh e a t m a p s (t o p r o w ) . W i t hh i g h e r g L , h i g h e r c i n c r e a s e s t h e fi r i n g r e g u l a r i t y f o r a w i d e rr a n g e o f fi r i n g r a t e s a nd a w i d e rr a n g e o f c . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r fi r i n g r e g u l a r i t y . upp l e m e n t a r y F i g u r e : V a r y i n g t h e a p li t ud e o f i nh i b i t o r y s y n a p s e s . W i t h g r e a t e r a m p li t ud e o f t h e i nh i b i t o r y s y n a p s e s ( A i , E q . ( )) , t h e s t a b ili s a t i o n o f t h e m e m b r a n e p o t e n t i a li s o b s e r v e d o n l y f o r h i g h e r v a l u e s o f c (t o p r o w ) . T h i s i s r e fl ec t e d i n t h e fi r i n g r e g u l a r i t y o f t h e D T - L I F m o d e l ( b o tt o m r o w ) , a n a l ogo u s l y t o t h e Supp l e m e n t a r y F i g u r e . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s ( f o r t h e t o p r o w ) o r h i g h e r / l o w e r fi r i n g r e g u l a r i t y ( f o r t h e b o tt o m r o w ) . upp l e m e n t a r y F i g u r e : C o n t i nu o u s c h a n g e o f C V i n t h e HH - D T m o d e l. L e f t : S a m e a s F i g . C , bu t f o r c ∈ { , . , . , . , . } . D a s h e d a ndd o tt e d v e r t i c a lli n e s w i t hp o i n t s i nd i c a t ee t h ee l o w ee s t v a l u ee
Neuronal Dy-namics (Cambridge University Press, 2019).[47] C. Teeter, R. Iyer, V. Menon, N. Gouwens, D. Feng,J. Berg, A. Szafer, N. Cain, H. Zeng, M. Hawrylycz,C. Koch, and S. Mihalas, Generalized leaky integrate-and-fire models classify multiple neuron types, Nat. Com-mun , 709 (2018).[48] J. Benda, L. Maler, and A. Longtin, Linear Versus Non-linear Signal Transmission in Neuron Models With Adap-tation Currents or Dynamic Thresholds, J. Neurophysiol , 2806 (2010).[49] A. Destexhe and D. Par´e, Impact of Network Activityon the Integrative Properties of Neocortical PyramidalNeurons In Vivo, J. Neurophysiol , 1531 (1999).[50] W. R. Softky and C. Koch, The highly irregular firing ofcortical cells is inconsistent with temporal integration ofrandom EPSPs, J. Neurosci , 334 (1993).[51] D. P. Heyman and M. J. Sobel, Stochastic Models inOperations Research: Stochastic Processes and Operat- ing Characteristics , Dover Books on Computer ScienceSeries (Dover Publications, 2004).[52] B. Lindner, Superposition of many independent spiketrains is generally not a Poisson process, Phys Rev E , 022901 (2006).[53] J. F. C. Kingman, Poisson Processes (Oxford Studies inProbability) (Clarendon Press, 1993).[54] D. J. Amit and M. V. Tsodyks, Effective neurons and at-tractor neural networks in cortical environment, Network , 121 (1992).[55] T. Toyoizumi, K. Aihara, and S.-i. Amari, Fisher Infor-mation for Spike-Based Population Decoding, Phys. Rev.Lett. , 098102 (2006).[56] L. Kostal, P. Lansky, and J.-P. Rospars, Neuronal cod-ing and spiking randomness, Eur. J. Neurosci , 2693(2007).[57] R. R. de Ruyter van Steveninck, Reproducibility andVariability in Neural Spike Trains, Science , 1805(1997).[58] S. P. Strong, R. Koberle, R. R. de Ruyter van Steveninck,and W. Bialek, Entropy and Information in Neural SpikeTrains, Phys. Rev. Lett. , 197 (1998).[59] P. H. E. Tiesinga, J. V. Jos´e, and T. J. Sejnowski, Com-parison of current-driven and conductance-driven neocor-tical model neurons with Hodgkin-Huxley voltage-gatedchannels, Phys Rev E , 8413 (2000).[60] M. J. Chacron, B. Lindner, and A. Longtin, Noise Shap-ing by Interval Correlations Increases Information Trans-fer, Phys. Rev. Lett. , 080601 (2004).[61] B. Lindner, M. J. Chacron, and A. Longtin, Integrate-and-fire neurons with threshold noise: A tractable modelof how interspike interval correlations affect neuronal sig-nal transmission, Phys Rev E , 021911 (2005).[62] F. Farkhooi, E. Muller, and M. P. Nawrot, Adaptationreduces variability of the neuronal population code, PhysRev E , 050905 (2011).[63] B. Lindner and A. Longtin, Comment on “Characteri-zation of Subthreshold Voltage Fluctuations in NeuronalMembranes,” by M. Rudolph and A. Destexhe, NeuralComput. , 1896 (2006).[64] D. Sterratt, B. Graham, A. Gillies, and D. Willshaw, Principles of Computational Modelling in Neuroscience (Cambridge University Press, 2011).[65] M. J. Chacron, A. Longtin, M. St-Hilaire, and L. Maler,Suprathreshold Stochastic Firing Dynamics with Mem-ory in P-Type Electroreceptors, Phys. Rev. Lett. , 1576(2000).[66] R. Kobayashi, Y. Tsubo, and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci. , 9 (2009).[67] W. Gerstner and R. Naud, How Good Are Neuron Mod-els?, Science , 379 (2009).[68] R. Kobayashi, S. Kurita, A. Kurth, K. Kitano,K. Mizuseki, M. Diesmann, B. J. Richmond, and S. Shi-nomoto, Reconstructing neuronal circuitry from parallelspike trains, Nat. Commun. , 4468 (2019).[69] R. Moreno-Bote and N. Parga, Membrane Potential andResponse Properties of Populations of Cortical Neuronsin the High Conductance State, Phys. Rev. Lett. ,088103 (2005).[70] R. Moreno-Bote and N. Parga, Auto- and Crosscorrel-ograms for the Spike Response of Leaky Integrate-and-Fire Neurons with Slow Synapses, Phys. Rev. Lett. , 028101 (2006).[71] R. Moreno-Bote, A. Renart, and N. Parga, Theory of In-put Spike Auto- and Cross-Correlations and Their Effecton the Response of Spiking Neurons, Neural Comput. ,1651 (2008).[72] R. Moreno-Bote and N. Parga, Response of Integrate-and-Fire Neurons to Noisy Inputs Filtered by Synapseswith Arbitrary Timescales: Firing Rate and Correlations,Neural Comput. , 1528 (2010). upp l e m e n t a r y F i g u r e : C o n s t r u c t i o n a nd v a li d a t i o n o f d y n a m i c t h r e s h o l d HH m o d e l. A - D : T h e b l u e li n e s r e p r e s e n tt h e a c t i v a t i o n ( A ) a nd i n a c t i v a t i o n ( B ) r a t e f un c t i o n s o f t h e ga t i n g v a r i a b l e h o f t h e n o n - a d a p t i n g HH m o d e l ( HH - ) . T h e o r a n g e li n e s r e p r e s e n tt h e m o d i fi e d r a t e f un c t i o n s f o r t h e HH - D T m o d e l. A s a n e ff ec t o f t h i s m o d i fi c a t i o n , t h e r ec o v e r y t i m e i n t h e HH - D T m o d e li s p r o l o n g e d ( C ) , w h il e t h e s t e a d y s t a t e a c t i v a t i o n h ∞ r e m a i n s r e l a t i v e l y un c h a n g e d ( D ) . E : C o m pu t a t i o n o f t h e t h r e s h o l d . I f n oa dd i t i o n a l p r e - s y n a p t i c s p i k e s a rr i v e a f t e r t = . m s , n o s p i k e i s o b s e r v e d ( g r ee n t r a ce ) . I f t h e p r e - s y n a p t i c s p i k e ss t o p a rr i v i n g ∆ t = . m s l a t e r , a s p i k e i s o b s e r v e d . T h e r e f o r e , w e d e fin e t h e t h r e s h o l d a s b e i n g r e a c h e d a t t = . m s . F - H : C o m p a r i s o n o f t h ec a l c u l a t e d t h r e s h o l d w i t h t h e p r ece d i n g i n t e r s p i k e i n t e r v a l ( I S I ) . W e v a li d a t e d t h a tt h e t h r e s h o l dd ec r e a s e s w i t h t i m e a f t e r a s p i k e i s fi r e db y e s t i m a t i n g t h e fi r i n g t h r e s h o l d i nd i v i du a ll y f o r e a c h s p i k e du r i n ga s i m u l a t i o n w i t h a s t a t i o n a r y i npu t , r e s u l t i n g i n a p o s t - s y n a p t i c fi r i n g r a t e o f a pp r o x i m a t e l y H z N e i t h e r t h e n o n - a d a p t i n g m o d e l ( HH - , p a n e l F ) n o r t h e m o d e l w i t h M - c u rr e n t a d a p t a t i o n ( HH - M , p a n e l G ) d i s p l a y ss i g n s o f s i g n i fi c a n t d e p e nd e n ce b e t w ee n t h e t h r e s h o l d a nd I S I . F o r t h e d y n a m i c t h r e s h o l d m o d e l ( HH - D T , p a n e l H ) w e o b s e r v e d a c l e a r d ec r e a s e i n t h e t h r e s h o l d w i t h i n c r e a s i n g du r a t i o n o f t h e p r ece d i n g I S I . T h e r e d li n e , a c t i n ga s a v i s u a l a i d , w a s o b t a i n e d b y a K NN r e g r e ss i o n . upp l e m e n t a r y F i g u r e : V a r y i n g t h e m e m b r a n e t i m e c o n s t a n t . T o p r o w : T h e m e m b r a n e t i m ec o n s t a n t i s v a r i e db y c h a n g i n g t h e l e a k c o ndu c t a n ce g L . W i t h g r e a t e r l e a k c o ndu c t a n ce , t h e m e m b r a n e p o t e n t i a l V f o ll o w s V e f m o r ec l o s e l y a nd t h e nh i g h e r c l e a d s t o l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s m o r e un i v e r s a ll y ( a pp r o x i m a t i o n o f σ V , c a l c u l a t e d f r o m E T A ,i s c o l o r c o d e d ) . T h e r e g i o n s w h e r e h i g h e r c d ec r e a s e s / i n c r e a s e s t h e m e m b r a n e p o t e n t i a l a r e s e p a r a t e db y t h e d o tt e d li n e . B o tt o m r o w : T h e s t r u c t u r e o f t h ec o n t o u r p l o t s i nd i c a t i n g h o w fi r i n g r e g u l a r i t y ( C V , c o l o r c o d e d ) c h a n g e s w i t h c a nd P S F R f o ll o w s t h e s t r u c t u r e o f t h e m e m b r a n e p o t e n t i a l flu c t u a t i o nh e a t m a p s (t o p r o w ) . W i t hh i g h e r g L , h i g h e r c i n c r e a s e s t h e fi r i n g r e g u l a r i t y f o r a w i d e rr a n g e o f fi r i n g r a t e s a nd a w i d e rr a n g e o f c . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r fi r i n g r e g u l a r i t y . upp l e m e n t a r y F i g u r e : V a r y i n g t h e a p li t ud e o f i nh i b i t o r y s y n a p s e s . W i t h g r e a t e r a m p li t ud e o f t h e i nh i b i t o r y s y n a p s e s ( A i , E q . ( )) , t h e s t a b ili s a t i o n o f t h e m e m b r a n e p o t e n t i a li s o b s e r v e d o n l y f o r h i g h e r v a l u e s o f c (t o p r o w ) . T h i s i s r e fl ec t e d i n t h e fi r i n g r e g u l a r i t y o f t h e D T - L I F m o d e l ( b o tt o m r o w ) , a n a l ogo u s l y t o t h e Supp l e m e n t a r y F i g u r e . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s ( f o r t h e t o p r o w ) o r h i g h e r / l o w e r fi r i n g r e g u l a r i t y ( f o r t h e b o tt o m r o w ) . upp l e m e n t a r y F i g u r e : C o n t i nu o u s c h a n g e o f C V i n t h e HH - D T m o d e l. L e f t : S a m e a s F i g . C , bu t f o r c ∈ { , . , . , . , . } . D a s h e d a ndd o tt e d v e r t i c a lli n e s w i t hp o i n t s i nd i c a t ee t h ee l o w ee s t v a l u ee o f C V f o r ee
Neuronal Dy-namics (Cambridge University Press, 2019).[47] C. Teeter, R. Iyer, V. Menon, N. Gouwens, D. Feng,J. Berg, A. Szafer, N. Cain, H. Zeng, M. Hawrylycz,C. Koch, and S. Mihalas, Generalized leaky integrate-and-fire models classify multiple neuron types, Nat. Com-mun , 709 (2018).[48] J. Benda, L. Maler, and A. Longtin, Linear Versus Non-linear Signal Transmission in Neuron Models With Adap-tation Currents or Dynamic Thresholds, J. Neurophysiol , 2806 (2010).[49] A. Destexhe and D. Par´e, Impact of Network Activityon the Integrative Properties of Neocortical PyramidalNeurons In Vivo, J. Neurophysiol , 1531 (1999).[50] W. R. Softky and C. Koch, The highly irregular firing ofcortical cells is inconsistent with temporal integration ofrandom EPSPs, J. Neurosci , 334 (1993).[51] D. P. Heyman and M. J. Sobel, Stochastic Models inOperations Research: Stochastic Processes and Operat- ing Characteristics , Dover Books on Computer ScienceSeries (Dover Publications, 2004).[52] B. Lindner, Superposition of many independent spiketrains is generally not a Poisson process, Phys Rev E , 022901 (2006).[53] J. F. C. Kingman, Poisson Processes (Oxford Studies inProbability) (Clarendon Press, 1993).[54] D. J. Amit and M. V. Tsodyks, Effective neurons and at-tractor neural networks in cortical environment, Network , 121 (1992).[55] T. Toyoizumi, K. Aihara, and S.-i. Amari, Fisher Infor-mation for Spike-Based Population Decoding, Phys. Rev.Lett. , 098102 (2006).[56] L. Kostal, P. Lansky, and J.-P. Rospars, Neuronal cod-ing and spiking randomness, Eur. J. Neurosci , 2693(2007).[57] R. R. de Ruyter van Steveninck, Reproducibility andVariability in Neural Spike Trains, Science , 1805(1997).[58] S. P. Strong, R. Koberle, R. R. de Ruyter van Steveninck,and W. Bialek, Entropy and Information in Neural SpikeTrains, Phys. Rev. Lett. , 197 (1998).[59] P. H. E. Tiesinga, J. V. Jos´e, and T. J. Sejnowski, Com-parison of current-driven and conductance-driven neocor-tical model neurons with Hodgkin-Huxley voltage-gatedchannels, Phys Rev E , 8413 (2000).[60] M. J. Chacron, B. Lindner, and A. Longtin, Noise Shap-ing by Interval Correlations Increases Information Trans-fer, Phys. Rev. Lett. , 080601 (2004).[61] B. Lindner, M. J. Chacron, and A. Longtin, Integrate-and-fire neurons with threshold noise: A tractable modelof how interspike interval correlations affect neuronal sig-nal transmission, Phys Rev E , 021911 (2005).[62] F. Farkhooi, E. Muller, and M. P. Nawrot, Adaptationreduces variability of the neuronal population code, PhysRev E , 050905 (2011).[63] B. Lindner and A. Longtin, Comment on “Characteri-zation of Subthreshold Voltage Fluctuations in NeuronalMembranes,” by M. Rudolph and A. Destexhe, NeuralComput. , 1896 (2006).[64] D. Sterratt, B. Graham, A. Gillies, and D. Willshaw, Principles of Computational Modelling in Neuroscience (Cambridge University Press, 2011).[65] M. J. Chacron, A. Longtin, M. St-Hilaire, and L. Maler,Suprathreshold Stochastic Firing Dynamics with Mem-ory in P-Type Electroreceptors, Phys. Rev. Lett. , 1576(2000).[66] R. Kobayashi, Y. Tsubo, and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci. , 9 (2009).[67] W. Gerstner and R. Naud, How Good Are Neuron Mod-els?, Science , 379 (2009).[68] R. Kobayashi, S. Kurita, A. Kurth, K. Kitano,K. Mizuseki, M. Diesmann, B. J. Richmond, and S. Shi-nomoto, Reconstructing neuronal circuitry from parallelspike trains, Nat. Commun. , 4468 (2019).[69] R. Moreno-Bote and N. Parga, Membrane Potential andResponse Properties of Populations of Cortical Neuronsin the High Conductance State, Phys. Rev. Lett. ,088103 (2005).[70] R. Moreno-Bote and N. Parga, Auto- and Crosscorrel-ograms for the Spike Response of Leaky Integrate-and-Fire Neurons with Slow Synapses, Phys. Rev. Lett. , 028101 (2006).[71] R. Moreno-Bote, A. Renart, and N. Parga, Theory of In-put Spike Auto- and Cross-Correlations and Their Effecton the Response of Spiking Neurons, Neural Comput. ,1651 (2008).[72] R. Moreno-Bote and N. Parga, Response of Integrate-and-Fire Neurons to Noisy Inputs Filtered by Synapseswith Arbitrary Timescales: Firing Rate and Correlations,Neural Comput. , 1528 (2010). upp l e m e n t a r y F i g u r e : C o n s t r u c t i o n a nd v a li d a t i o n o f d y n a m i c t h r e s h o l d HH m o d e l. A - D : T h e b l u e li n e s r e p r e s e n tt h e a c t i v a t i o n ( A ) a nd i n a c t i v a t i o n ( B ) r a t e f un c t i o n s o f t h e ga t i n g v a r i a b l e h o f t h e n o n - a d a p t i n g HH m o d e l ( HH - ) . T h e o r a n g e li n e s r e p r e s e n tt h e m o d i fi e d r a t e f un c t i o n s f o r t h e HH - D T m o d e l. A s a n e ff ec t o f t h i s m o d i fi c a t i o n , t h e r ec o v e r y t i m e i n t h e HH - D T m o d e li s p r o l o n g e d ( C ) , w h il e t h e s t e a d y s t a t e a c t i v a t i o n h ∞ r e m a i n s r e l a t i v e l y un c h a n g e d ( D ) . E : C o m pu t a t i o n o f t h e t h r e s h o l d . I f n oa dd i t i o n a l p r e - s y n a p t i c s p i k e s a rr i v e a f t e r t = . m s , n o s p i k e i s o b s e r v e d ( g r ee n t r a ce ) . I f t h e p r e - s y n a p t i c s p i k e ss t o p a rr i v i n g ∆ t = . m s l a t e r , a s p i k e i s o b s e r v e d . T h e r e f o r e , w e d e fin e t h e t h r e s h o l d a s b e i n g r e a c h e d a t t = . m s . F - H : C o m p a r i s o n o f t h ec a l c u l a t e d t h r e s h o l d w i t h t h e p r ece d i n g i n t e r s p i k e i n t e r v a l ( I S I ) . W e v a li d a t e d t h a tt h e t h r e s h o l dd ec r e a s e s w i t h t i m e a f t e r a s p i k e i s fi r e db y e s t i m a t i n g t h e fi r i n g t h r e s h o l d i nd i v i du a ll y f o r e a c h s p i k e du r i n ga s i m u l a t i o n w i t h a s t a t i o n a r y i npu t , r e s u l t i n g i n a p o s t - s y n a p t i c fi r i n g r a t e o f a pp r o x i m a t e l y H z N e i t h e r t h e n o n - a d a p t i n g m o d e l ( HH - , p a n e l F ) n o r t h e m o d e l w i t h M - c u rr e n t a d a p t a t i o n ( HH - M , p a n e l G ) d i s p l a y ss i g n s o f s i g n i fi c a n t d e p e nd e n ce b e t w ee n t h e t h r e s h o l d a nd I S I . F o r t h e d y n a m i c t h r e s h o l d m o d e l ( HH - D T , p a n e l H ) w e o b s e r v e d a c l e a r d ec r e a s e i n t h e t h r e s h o l d w i t h i n c r e a s i n g du r a t i o n o f t h e p r ece d i n g I S I . T h e r e d li n e , a c t i n ga s a v i s u a l a i d , w a s o b t a i n e d b y a K NN r e g r e ss i o n . upp l e m e n t a r y F i g u r e : V a r y i n g t h e m e m b r a n e t i m e c o n s t a n t . T o p r o w : T h e m e m b r a n e t i m ec o n s t a n t i s v a r i e db y c h a n g i n g t h e l e a k c o ndu c t a n ce g L . W i t h g r e a t e r l e a k c o ndu c t a n ce , t h e m e m b r a n e p o t e n t i a l V f o ll o w s V e f m o r ec l o s e l y a nd t h e nh i g h e r c l e a d s t o l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s m o r e un i v e r s a ll y ( a pp r o x i m a t i o n o f σ V , c a l c u l a t e d f r o m E T A ,i s c o l o r c o d e d ) . T h e r e g i o n s w h e r e h i g h e r c d ec r e a s e s / i n c r e a s e s t h e m e m b r a n e p o t e n t i a l a r e s e p a r a t e db y t h e d o tt e d li n e . B o tt o m r o w : T h e s t r u c t u r e o f t h ec o n t o u r p l o t s i nd i c a t i n g h o w fi r i n g r e g u l a r i t y ( C V , c o l o r c o d e d ) c h a n g e s w i t h c a nd P S F R f o ll o w s t h e s t r u c t u r e o f t h e m e m b r a n e p o t e n t i a l flu c t u a t i o nh e a t m a p s (t o p r o w ) . W i t hh i g h e r g L , h i g h e r c i n c r e a s e s t h e fi r i n g r e g u l a r i t y f o r a w i d e rr a n g e o f fi r i n g r a t e s a nd a w i d e rr a n g e o f c . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r fi r i n g r e g u l a r i t y . upp l e m e n t a r y F i g u r e : V a r y i n g t h e a p li t ud e o f i nh i b i t o r y s y n a p s e s . W i t h g r e a t e r a m p li t ud e o f t h e i nh i b i t o r y s y n a p s e s ( A i , E q . ( )) , t h e s t a b ili s a t i o n o f t h e m e m b r a n e p o t e n t i a li s o b s e r v e d o n l y f o r h i g h e r v a l u e s o f c (t o p r o w ) . T h i s i s r e fl ec t e d i n t h e fi r i n g r e g u l a r i t y o f t h e D T - L I F m o d e l ( b o tt o m r o w ) , a n a l ogo u s l y t o t h e Supp l e m e n t a r y F i g u r e . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s ( f o r t h e t o p r o w ) o r h i g h e r / l o w e r fi r i n g r e g u l a r i t y ( f o r t h e b o tt o m r o w ) . upp l e m e n t a r y F i g u r e : C o n t i nu o u s c h a n g e o f C V i n t h e HH - D T m o d e l. L e f t : S a m e a s F i g . C , bu t f o r c ∈ { , . , . , . , . } . D a s h e d a ndd o tt e d v e r t i c a lli n e s w i t hp o i n t s i nd i c a t ee t h ee l o w ee s t v a l u ee o f C V f o r ee a c h c . T h ee
Neuronal Dy-namics (Cambridge University Press, 2019).[47] C. Teeter, R. Iyer, V. Menon, N. Gouwens, D. Feng,J. Berg, A. Szafer, N. Cain, H. Zeng, M. Hawrylycz,C. Koch, and S. Mihalas, Generalized leaky integrate-and-fire models classify multiple neuron types, Nat. Com-mun , 709 (2018).[48] J. Benda, L. Maler, and A. Longtin, Linear Versus Non-linear Signal Transmission in Neuron Models With Adap-tation Currents or Dynamic Thresholds, J. Neurophysiol , 2806 (2010).[49] A. Destexhe and D. Par´e, Impact of Network Activityon the Integrative Properties of Neocortical PyramidalNeurons In Vivo, J. Neurophysiol , 1531 (1999).[50] W. R. Softky and C. Koch, The highly irregular firing ofcortical cells is inconsistent with temporal integration ofrandom EPSPs, J. Neurosci , 334 (1993).[51] D. P. Heyman and M. J. Sobel, Stochastic Models inOperations Research: Stochastic Processes and Operat- ing Characteristics , Dover Books on Computer ScienceSeries (Dover Publications, 2004).[52] B. Lindner, Superposition of many independent spiketrains is generally not a Poisson process, Phys Rev E , 022901 (2006).[53] J. F. C. Kingman, Poisson Processes (Oxford Studies inProbability) (Clarendon Press, 1993).[54] D. J. Amit and M. V. Tsodyks, Effective neurons and at-tractor neural networks in cortical environment, Network , 121 (1992).[55] T. Toyoizumi, K. Aihara, and S.-i. Amari, Fisher Infor-mation for Spike-Based Population Decoding, Phys. Rev.Lett. , 098102 (2006).[56] L. Kostal, P. Lansky, and J.-P. Rospars, Neuronal cod-ing and spiking randomness, Eur. J. Neurosci , 2693(2007).[57] R. R. de Ruyter van Steveninck, Reproducibility andVariability in Neural Spike Trains, Science , 1805(1997).[58] S. P. Strong, R. Koberle, R. R. de Ruyter van Steveninck,and W. Bialek, Entropy and Information in Neural SpikeTrains, Phys. Rev. Lett. , 197 (1998).[59] P. H. E. Tiesinga, J. V. Jos´e, and T. J. Sejnowski, Com-parison of current-driven and conductance-driven neocor-tical model neurons with Hodgkin-Huxley voltage-gatedchannels, Phys Rev E , 8413 (2000).[60] M. J. Chacron, B. Lindner, and A. Longtin, Noise Shap-ing by Interval Correlations Increases Information Trans-fer, Phys. Rev. Lett. , 080601 (2004).[61] B. Lindner, M. J. Chacron, and A. Longtin, Integrate-and-fire neurons with threshold noise: A tractable modelof how interspike interval correlations affect neuronal sig-nal transmission, Phys Rev E , 021911 (2005).[62] F. Farkhooi, E. Muller, and M. P. Nawrot, Adaptationreduces variability of the neuronal population code, PhysRev E , 050905 (2011).[63] B. Lindner and A. Longtin, Comment on “Characteri-zation of Subthreshold Voltage Fluctuations in NeuronalMembranes,” by M. Rudolph and A. Destexhe, NeuralComput. , 1896 (2006).[64] D. Sterratt, B. Graham, A. Gillies, and D. Willshaw, Principles of Computational Modelling in Neuroscience (Cambridge University Press, 2011).[65] M. J. Chacron, A. Longtin, M. St-Hilaire, and L. Maler,Suprathreshold Stochastic Firing Dynamics with Mem-ory in P-Type Electroreceptors, Phys. Rev. Lett. , 1576(2000).[66] R. Kobayashi, Y. Tsubo, and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci. , 9 (2009).[67] W. Gerstner and R. Naud, How Good Are Neuron Mod-els?, Science , 379 (2009).[68] R. Kobayashi, S. Kurita, A. Kurth, K. Kitano,K. Mizuseki, M. Diesmann, B. J. Richmond, and S. Shi-nomoto, Reconstructing neuronal circuitry from parallelspike trains, Nat. Commun. , 4468 (2019).[69] R. Moreno-Bote and N. Parga, Membrane Potential andResponse Properties of Populations of Cortical Neuronsin the High Conductance State, Phys. Rev. Lett. ,088103 (2005).[70] R. Moreno-Bote and N. Parga, Auto- and Crosscorrel-ograms for the Spike Response of Leaky Integrate-and-Fire Neurons with Slow Synapses, Phys. Rev. Lett. , 028101 (2006).[71] R. Moreno-Bote, A. Renart, and N. Parga, Theory of In-put Spike Auto- and Cross-Correlations and Their Effecton the Response of Spiking Neurons, Neural Comput. ,1651 (2008).[72] R. Moreno-Bote and N. Parga, Response of Integrate-and-Fire Neurons to Noisy Inputs Filtered by Synapseswith Arbitrary Timescales: Firing Rate and Correlations,Neural Comput. , 1528 (2010). upp l e m e n t a r y F i g u r e : C o n s t r u c t i o n a nd v a li d a t i o n o f d y n a m i c t h r e s h o l d HH m o d e l. A - D : T h e b l u e li n e s r e p r e s e n tt h e a c t i v a t i o n ( A ) a nd i n a c t i v a t i o n ( B ) r a t e f un c t i o n s o f t h e ga t i n g v a r i a b l e h o f t h e n o n - a d a p t i n g HH m o d e l ( HH - ) . T h e o r a n g e li n e s r e p r e s e n tt h e m o d i fi e d r a t e f un c t i o n s f o r t h e HH - D T m o d e l. A s a n e ff ec t o f t h i s m o d i fi c a t i o n , t h e r ec o v e r y t i m e i n t h e HH - D T m o d e li s p r o l o n g e d ( C ) , w h il e t h e s t e a d y s t a t e a c t i v a t i o n h ∞ r e m a i n s r e l a t i v e l y un c h a n g e d ( D ) . E : C o m pu t a t i o n o f t h e t h r e s h o l d . I f n oa dd i t i o n a l p r e - s y n a p t i c s p i k e s a rr i v e a f t e r t = . m s , n o s p i k e i s o b s e r v e d ( g r ee n t r a ce ) . I f t h e p r e - s y n a p t i c s p i k e ss t o p a rr i v i n g ∆ t = . m s l a t e r , a s p i k e i s o b s e r v e d . T h e r e f o r e , w e d e fin e t h e t h r e s h o l d a s b e i n g r e a c h e d a t t = . m s . F - H : C o m p a r i s o n o f t h ec a l c u l a t e d t h r e s h o l d w i t h t h e p r ece d i n g i n t e r s p i k e i n t e r v a l ( I S I ) . W e v a li d a t e d t h a tt h e t h r e s h o l dd ec r e a s e s w i t h t i m e a f t e r a s p i k e i s fi r e db y e s t i m a t i n g t h e fi r i n g t h r e s h o l d i nd i v i du a ll y f o r e a c h s p i k e du r i n ga s i m u l a t i o n w i t h a s t a t i o n a r y i npu t , r e s u l t i n g i n a p o s t - s y n a p t i c fi r i n g r a t e o f a pp r o x i m a t e l y H z N e i t h e r t h e n o n - a d a p t i n g m o d e l ( HH - , p a n e l F ) n o r t h e m o d e l w i t h M - c u rr e n t a d a p t a t i o n ( HH - M , p a n e l G ) d i s p l a y ss i g n s o f s i g n i fi c a n t d e p e nd e n ce b e t w ee n t h e t h r e s h o l d a nd I S I . F o r t h e d y n a m i c t h r e s h o l d m o d e l ( HH - D T , p a n e l H ) w e o b s e r v e d a c l e a r d ec r e a s e i n t h e t h r e s h o l d w i t h i n c r e a s i n g du r a t i o n o f t h e p r ece d i n g I S I . T h e r e d li n e , a c t i n ga s a v i s u a l a i d , w a s o b t a i n e d b y a K NN r e g r e ss i o n . upp l e m e n t a r y F i g u r e : V a r y i n g t h e m e m b r a n e t i m e c o n s t a n t . T o p r o w : T h e m e m b r a n e t i m ec o n s t a n t i s v a r i e db y c h a n g i n g t h e l e a k c o ndu c t a n ce g L . W i t h g r e a t e r l e a k c o ndu c t a n ce , t h e m e m b r a n e p o t e n t i a l V f o ll o w s V e f m o r ec l o s e l y a nd t h e nh i g h e r c l e a d s t o l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s m o r e un i v e r s a ll y ( a pp r o x i m a t i o n o f σ V , c a l c u l a t e d f r o m E T A ,i s c o l o r c o d e d ) . T h e r e g i o n s w h e r e h i g h e r c d ec r e a s e s / i n c r e a s e s t h e m e m b r a n e p o t e n t i a l a r e s e p a r a t e db y t h e d o tt e d li n e . B o tt o m r o w : T h e s t r u c t u r e o f t h ec o n t o u r p l o t s i nd i c a t i n g h o w fi r i n g r e g u l a r i t y ( C V , c o l o r c o d e d ) c h a n g e s w i t h c a nd P S F R f o ll o w s t h e s t r u c t u r e o f t h e m e m b r a n e p o t e n t i a l flu c t u a t i o nh e a t m a p s (t o p r o w ) . W i t hh i g h e r g L , h i g h e r c i n c r e a s e s t h e fi r i n g r e g u l a r i t y f o r a w i d e rr a n g e o f fi r i n g r a t e s a nd a w i d e rr a n g e o f c . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r fi r i n g r e g u l a r i t y . upp l e m e n t a r y F i g u r e : V a r y i n g t h e a p li t ud e o f i nh i b i t o r y s y n a p s e s . W i t h g r e a t e r a m p li t ud e o f t h e i nh i b i t o r y s y n a p s e s ( A i , E q . ( )) , t h e s t a b ili s a t i o n o f t h e m e m b r a n e p o t e n t i a li s o b s e r v e d o n l y f o r h i g h e r v a l u e s o f c (t o p r o w ) . T h i s i s r e fl ec t e d i n t h e fi r i n g r e g u l a r i t y o f t h e D T - L I F m o d e l ( b o tt o m r o w ) , a n a l ogo u s l y t o t h e Supp l e m e n t a r y F i g u r e . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s ( f o r t h e t o p r o w ) o r h i g h e r / l o w e r fi r i n g r e g u l a r i t y ( f o r t h e b o tt o m r o w ) . upp l e m e n t a r y F i g u r e : C o n t i nu o u s c h a n g e o f C V i n t h e HH - D T m o d e l. L e f t : S a m e a s F i g . C , bu t f o r c ∈ { , . , . , . , . } . D a s h e d a ndd o tt e d v e r t i c a lli n e s w i t hp o i n t s i nd i c a t ee t h ee l o w ee s t v a l u ee o f C V f o r ee a c h c . T h ee s ee
Neuronal Dy-namics (Cambridge University Press, 2019).[47] C. Teeter, R. Iyer, V. Menon, N. Gouwens, D. Feng,J. Berg, A. Szafer, N. Cain, H. Zeng, M. Hawrylycz,C. Koch, and S. Mihalas, Generalized leaky integrate-and-fire models classify multiple neuron types, Nat. Com-mun , 709 (2018).[48] J. Benda, L. Maler, and A. Longtin, Linear Versus Non-linear Signal Transmission in Neuron Models With Adap-tation Currents or Dynamic Thresholds, J. Neurophysiol , 2806 (2010).[49] A. Destexhe and D. Par´e, Impact of Network Activityon the Integrative Properties of Neocortical PyramidalNeurons In Vivo, J. Neurophysiol , 1531 (1999).[50] W. R. Softky and C. Koch, The highly irregular firing ofcortical cells is inconsistent with temporal integration ofrandom EPSPs, J. Neurosci , 334 (1993).[51] D. P. Heyman and M. J. Sobel, Stochastic Models inOperations Research: Stochastic Processes and Operat- ing Characteristics , Dover Books on Computer ScienceSeries (Dover Publications, 2004).[52] B. Lindner, Superposition of many independent spiketrains is generally not a Poisson process, Phys Rev E , 022901 (2006).[53] J. F. C. Kingman, Poisson Processes (Oxford Studies inProbability) (Clarendon Press, 1993).[54] D. J. Amit and M. V. Tsodyks, Effective neurons and at-tractor neural networks in cortical environment, Network , 121 (1992).[55] T. Toyoizumi, K. Aihara, and S.-i. Amari, Fisher Infor-mation for Spike-Based Population Decoding, Phys. Rev.Lett. , 098102 (2006).[56] L. Kostal, P. Lansky, and J.-P. Rospars, Neuronal cod-ing and spiking randomness, Eur. J. Neurosci , 2693(2007).[57] R. R. de Ruyter van Steveninck, Reproducibility andVariability in Neural Spike Trains, Science , 1805(1997).[58] S. P. Strong, R. Koberle, R. R. de Ruyter van Steveninck,and W. Bialek, Entropy and Information in Neural SpikeTrains, Phys. Rev. Lett. , 197 (1998).[59] P. H. E. Tiesinga, J. V. Jos´e, and T. J. Sejnowski, Com-parison of current-driven and conductance-driven neocor-tical model neurons with Hodgkin-Huxley voltage-gatedchannels, Phys Rev E , 8413 (2000).[60] M. J. Chacron, B. Lindner, and A. Longtin, Noise Shap-ing by Interval Correlations Increases Information Trans-fer, Phys. Rev. Lett. , 080601 (2004).[61] B. Lindner, M. J. Chacron, and A. Longtin, Integrate-and-fire neurons with threshold noise: A tractable modelof how interspike interval correlations affect neuronal sig-nal transmission, Phys Rev E , 021911 (2005).[62] F. Farkhooi, E. Muller, and M. P. Nawrot, Adaptationreduces variability of the neuronal population code, PhysRev E , 050905 (2011).[63] B. Lindner and A. Longtin, Comment on “Characteri-zation of Subthreshold Voltage Fluctuations in NeuronalMembranes,” by M. Rudolph and A. Destexhe, NeuralComput. , 1896 (2006).[64] D. Sterratt, B. Graham, A. Gillies, and D. Willshaw, Principles of Computational Modelling in Neuroscience (Cambridge University Press, 2011).[65] M. J. Chacron, A. Longtin, M. St-Hilaire, and L. Maler,Suprathreshold Stochastic Firing Dynamics with Mem-ory in P-Type Electroreceptors, Phys. Rev. Lett. , 1576(2000).[66] R. Kobayashi, Y. Tsubo, and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci. , 9 (2009).[67] W. Gerstner and R. Naud, How Good Are Neuron Mod-els?, Science , 379 (2009).[68] R. Kobayashi, S. Kurita, A. Kurth, K. Kitano,K. Mizuseki, M. Diesmann, B. J. Richmond, and S. Shi-nomoto, Reconstructing neuronal circuitry from parallelspike trains, Nat. Commun. , 4468 (2019).[69] R. Moreno-Bote and N. Parga, Membrane Potential andResponse Properties of Populations of Cortical Neuronsin the High Conductance State, Phys. Rev. Lett. ,088103 (2005).[70] R. Moreno-Bote and N. Parga, Auto- and Crosscorrel-ograms for the Spike Response of Leaky Integrate-and-Fire Neurons with Slow Synapses, Phys. Rev. Lett. , 028101 (2006).[71] R. Moreno-Bote, A. Renart, and N. Parga, Theory of In-put Spike Auto- and Cross-Correlations and Their Effecton the Response of Spiking Neurons, Neural Comput. ,1651 (2008).[72] R. Moreno-Bote and N. Parga, Response of Integrate-and-Fire Neurons to Noisy Inputs Filtered by Synapseswith Arbitrary Timescales: Firing Rate and Correlations,Neural Comput. , 1528 (2010). upp l e m e n t a r y F i g u r e : C o n s t r u c t i o n a nd v a li d a t i o n o f d y n a m i c t h r e s h o l d HH m o d e l. A - D : T h e b l u e li n e s r e p r e s e n tt h e a c t i v a t i o n ( A ) a nd i n a c t i v a t i o n ( B ) r a t e f un c t i o n s o f t h e ga t i n g v a r i a b l e h o f t h e n o n - a d a p t i n g HH m o d e l ( HH - ) . T h e o r a n g e li n e s r e p r e s e n tt h e m o d i fi e d r a t e f un c t i o n s f o r t h e HH - D T m o d e l. A s a n e ff ec t o f t h i s m o d i fi c a t i o n , t h e r ec o v e r y t i m e i n t h e HH - D T m o d e li s p r o l o n g e d ( C ) , w h il e t h e s t e a d y s t a t e a c t i v a t i o n h ∞ r e m a i n s r e l a t i v e l y un c h a n g e d ( D ) . E : C o m pu t a t i o n o f t h e t h r e s h o l d . I f n oa dd i t i o n a l p r e - s y n a p t i c s p i k e s a rr i v e a f t e r t = . m s , n o s p i k e i s o b s e r v e d ( g r ee n t r a ce ) . I f t h e p r e - s y n a p t i c s p i k e ss t o p a rr i v i n g ∆ t = . m s l a t e r , a s p i k e i s o b s e r v e d . T h e r e f o r e , w e d e fin e t h e t h r e s h o l d a s b e i n g r e a c h e d a t t = . m s . F - H : C o m p a r i s o n o f t h ec a l c u l a t e d t h r e s h o l d w i t h t h e p r ece d i n g i n t e r s p i k e i n t e r v a l ( I S I ) . W e v a li d a t e d t h a tt h e t h r e s h o l dd ec r e a s e s w i t h t i m e a f t e r a s p i k e i s fi r e db y e s t i m a t i n g t h e fi r i n g t h r e s h o l d i nd i v i du a ll y f o r e a c h s p i k e du r i n ga s i m u l a t i o n w i t h a s t a t i o n a r y i npu t , r e s u l t i n g i n a p o s t - s y n a p t i c fi r i n g r a t e o f a pp r o x i m a t e l y H z N e i t h e r t h e n o n - a d a p t i n g m o d e l ( HH - , p a n e l F ) n o r t h e m o d e l w i t h M - c u rr e n t a d a p t a t i o n ( HH - M , p a n e l G ) d i s p l a y ss i g n s o f s i g n i fi c a n t d e p e nd e n ce b e t w ee n t h e t h r e s h o l d a nd I S I . F o r t h e d y n a m i c t h r e s h o l d m o d e l ( HH - D T , p a n e l H ) w e o b s e r v e d a c l e a r d ec r e a s e i n t h e t h r e s h o l d w i t h i n c r e a s i n g du r a t i o n o f t h e p r ece d i n g I S I . T h e r e d li n e , a c t i n ga s a v i s u a l a i d , w a s o b t a i n e d b y a K NN r e g r e ss i o n . upp l e m e n t a r y F i g u r e : V a r y i n g t h e m e m b r a n e t i m e c o n s t a n t . T o p r o w : T h e m e m b r a n e t i m ec o n s t a n t i s v a r i e db y c h a n g i n g t h e l e a k c o ndu c t a n ce g L . W i t h g r e a t e r l e a k c o ndu c t a n ce , t h e m e m b r a n e p o t e n t i a l V f o ll o w s V e f m o r ec l o s e l y a nd t h e nh i g h e r c l e a d s t o l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s m o r e un i v e r s a ll y ( a pp r o x i m a t i o n o f σ V , c a l c u l a t e d f r o m E T A ,i s c o l o r c o d e d ) . T h e r e g i o n s w h e r e h i g h e r c d ec r e a s e s / i n c r e a s e s t h e m e m b r a n e p o t e n t i a l a r e s e p a r a t e db y t h e d o tt e d li n e . B o tt o m r o w : T h e s t r u c t u r e o f t h ec o n t o u r p l o t s i nd i c a t i n g h o w fi r i n g r e g u l a r i t y ( C V , c o l o r c o d e d ) c h a n g e s w i t h c a nd P S F R f o ll o w s t h e s t r u c t u r e o f t h e m e m b r a n e p o t e n t i a l flu c t u a t i o nh e a t m a p s (t o p r o w ) . W i t hh i g h e r g L , h i g h e r c i n c r e a s e s t h e fi r i n g r e g u l a r i t y f o r a w i d e rr a n g e o f fi r i n g r a t e s a nd a w i d e rr a n g e o f c . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r fi r i n g r e g u l a r i t y . upp l e m e n t a r y F i g u r e : V a r y i n g t h e a p li t ud e o f i nh i b i t o r y s y n a p s e s . W i t h g r e a t e r a m p li t ud e o f t h e i nh i b i t o r y s y n a p s e s ( A i , E q . ( )) , t h e s t a b ili s a t i o n o f t h e m e m b r a n e p o t e n t i a li s o b s e r v e d o n l y f o r h i g h e r v a l u e s o f c (t o p r o w ) . T h i s i s r e fl ec t e d i n t h e fi r i n g r e g u l a r i t y o f t h e D T - L I F m o d e l ( b o tt o m r o w ) , a n a l ogo u s l y t o t h e Supp l e m e n t a r y F i g u r e . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s ( f o r t h e t o p r o w ) o r h i g h e r / l o w e r fi r i n g r e g u l a r i t y ( f o r t h e b o tt o m r o w ) . upp l e m e n t a r y F i g u r e : C o n t i nu o u s c h a n g e o f C V i n t h e HH - D T m o d e l. L e f t : S a m e a s F i g . C , bu t f o r c ∈ { , . , . , . , . } . D a s h e d a ndd o tt e d v e r t i c a lli n e s w i t hp o i n t s i nd i c a t ee t h ee l o w ee s t v a l u ee o f C V f o r ee a c h c . T h ee s ee a r ee
Neuronal Dy-namics (Cambridge University Press, 2019).[47] C. Teeter, R. Iyer, V. Menon, N. Gouwens, D. Feng,J. Berg, A. Szafer, N. Cain, H. Zeng, M. Hawrylycz,C. Koch, and S. Mihalas, Generalized leaky integrate-and-fire models classify multiple neuron types, Nat. Com-mun , 709 (2018).[48] J. Benda, L. Maler, and A. Longtin, Linear Versus Non-linear Signal Transmission in Neuron Models With Adap-tation Currents or Dynamic Thresholds, J. Neurophysiol , 2806 (2010).[49] A. Destexhe and D. Par´e, Impact of Network Activityon the Integrative Properties of Neocortical PyramidalNeurons In Vivo, J. Neurophysiol , 1531 (1999).[50] W. R. Softky and C. Koch, The highly irregular firing ofcortical cells is inconsistent with temporal integration ofrandom EPSPs, J. Neurosci , 334 (1993).[51] D. P. Heyman and M. J. Sobel, Stochastic Models inOperations Research: Stochastic Processes and Operat- ing Characteristics , Dover Books on Computer ScienceSeries (Dover Publications, 2004).[52] B. Lindner, Superposition of many independent spiketrains is generally not a Poisson process, Phys Rev E , 022901 (2006).[53] J. F. C. Kingman, Poisson Processes (Oxford Studies inProbability) (Clarendon Press, 1993).[54] D. J. Amit and M. V. Tsodyks, Effective neurons and at-tractor neural networks in cortical environment, Network , 121 (1992).[55] T. Toyoizumi, K. Aihara, and S.-i. Amari, Fisher Infor-mation for Spike-Based Population Decoding, Phys. Rev.Lett. , 098102 (2006).[56] L. Kostal, P. Lansky, and J.-P. Rospars, Neuronal cod-ing and spiking randomness, Eur. J. Neurosci , 2693(2007).[57] R. R. de Ruyter van Steveninck, Reproducibility andVariability in Neural Spike Trains, Science , 1805(1997).[58] S. P. Strong, R. Koberle, R. R. de Ruyter van Steveninck,and W. Bialek, Entropy and Information in Neural SpikeTrains, Phys. Rev. Lett. , 197 (1998).[59] P. H. E. Tiesinga, J. V. Jos´e, and T. J. Sejnowski, Com-parison of current-driven and conductance-driven neocor-tical model neurons with Hodgkin-Huxley voltage-gatedchannels, Phys Rev E , 8413 (2000).[60] M. J. Chacron, B. Lindner, and A. Longtin, Noise Shap-ing by Interval Correlations Increases Information Trans-fer, Phys. Rev. Lett. , 080601 (2004).[61] B. Lindner, M. J. Chacron, and A. Longtin, Integrate-and-fire neurons with threshold noise: A tractable modelof how interspike interval correlations affect neuronal sig-nal transmission, Phys Rev E , 021911 (2005).[62] F. Farkhooi, E. Muller, and M. P. Nawrot, Adaptationreduces variability of the neuronal population code, PhysRev E , 050905 (2011).[63] B. Lindner and A. Longtin, Comment on “Characteri-zation of Subthreshold Voltage Fluctuations in NeuronalMembranes,” by M. Rudolph and A. Destexhe, NeuralComput. , 1896 (2006).[64] D. Sterratt, B. Graham, A. Gillies, and D. Willshaw, Principles of Computational Modelling in Neuroscience (Cambridge University Press, 2011).[65] M. J. Chacron, A. Longtin, M. St-Hilaire, and L. Maler,Suprathreshold Stochastic Firing Dynamics with Mem-ory in P-Type Electroreceptors, Phys. Rev. Lett. , 1576(2000).[66] R. Kobayashi, Y. Tsubo, and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci. , 9 (2009).[67] W. Gerstner and R. Naud, How Good Are Neuron Mod-els?, Science , 379 (2009).[68] R. Kobayashi, S. Kurita, A. Kurth, K. Kitano,K. Mizuseki, M. Diesmann, B. J. Richmond, and S. Shi-nomoto, Reconstructing neuronal circuitry from parallelspike trains, Nat. Commun. , 4468 (2019).[69] R. Moreno-Bote and N. Parga, Membrane Potential andResponse Properties of Populations of Cortical Neuronsin the High Conductance State, Phys. Rev. Lett. ,088103 (2005).[70] R. Moreno-Bote and N. Parga, Auto- and Crosscorrel-ograms for the Spike Response of Leaky Integrate-and-Fire Neurons with Slow Synapses, Phys. Rev. Lett. , 028101 (2006).[71] R. Moreno-Bote, A. Renart, and N. Parga, Theory of In-put Spike Auto- and Cross-Correlations and Their Effecton the Response of Spiking Neurons, Neural Comput. ,1651 (2008).[72] R. Moreno-Bote and N. Parga, Response of Integrate-and-Fire Neurons to Noisy Inputs Filtered by Synapseswith Arbitrary Timescales: Firing Rate and Correlations,Neural Comput. , 1528 (2010). upp l e m e n t a r y F i g u r e : C o n s t r u c t i o n a nd v a li d a t i o n o f d y n a m i c t h r e s h o l d HH m o d e l. A - D : T h e b l u e li n e s r e p r e s e n tt h e a c t i v a t i o n ( A ) a nd i n a c t i v a t i o n ( B ) r a t e f un c t i o n s o f t h e ga t i n g v a r i a b l e h o f t h e n o n - a d a p t i n g HH m o d e l ( HH - ) . T h e o r a n g e li n e s r e p r e s e n tt h e m o d i fi e d r a t e f un c t i o n s f o r t h e HH - D T m o d e l. A s a n e ff ec t o f t h i s m o d i fi c a t i o n , t h e r ec o v e r y t i m e i n t h e HH - D T m o d e li s p r o l o n g e d ( C ) , w h il e t h e s t e a d y s t a t e a c t i v a t i o n h ∞ r e m a i n s r e l a t i v e l y un c h a n g e d ( D ) . E : C o m pu t a t i o n o f t h e t h r e s h o l d . I f n oa dd i t i o n a l p r e - s y n a p t i c s p i k e s a rr i v e a f t e r t = . m s , n o s p i k e i s o b s e r v e d ( g r ee n t r a ce ) . I f t h e p r e - s y n a p t i c s p i k e ss t o p a rr i v i n g ∆ t = . m s l a t e r , a s p i k e i s o b s e r v e d . T h e r e f o r e , w e d e fin e t h e t h r e s h o l d a s b e i n g r e a c h e d a t t = . m s . F - H : C o m p a r i s o n o f t h ec a l c u l a t e d t h r e s h o l d w i t h t h e p r ece d i n g i n t e r s p i k e i n t e r v a l ( I S I ) . W e v a li d a t e d t h a tt h e t h r e s h o l dd ec r e a s e s w i t h t i m e a f t e r a s p i k e i s fi r e db y e s t i m a t i n g t h e fi r i n g t h r e s h o l d i nd i v i du a ll y f o r e a c h s p i k e du r i n ga s i m u l a t i o n w i t h a s t a t i o n a r y i npu t , r e s u l t i n g i n a p o s t - s y n a p t i c fi r i n g r a t e o f a pp r o x i m a t e l y H z N e i t h e r t h e n o n - a d a p t i n g m o d e l ( HH - , p a n e l F ) n o r t h e m o d e l w i t h M - c u rr e n t a d a p t a t i o n ( HH - M , p a n e l G ) d i s p l a y ss i g n s o f s i g n i fi c a n t d e p e nd e n ce b e t w ee n t h e t h r e s h o l d a nd I S I . F o r t h e d y n a m i c t h r e s h o l d m o d e l ( HH - D T , p a n e l H ) w e o b s e r v e d a c l e a r d ec r e a s e i n t h e t h r e s h o l d w i t h i n c r e a s i n g du r a t i o n o f t h e p r ece d i n g I S I . T h e r e d li n e , a c t i n ga s a v i s u a l a i d , w a s o b t a i n e d b y a K NN r e g r e ss i o n . upp l e m e n t a r y F i g u r e : V a r y i n g t h e m e m b r a n e t i m e c o n s t a n t . T o p r o w : T h e m e m b r a n e t i m ec o n s t a n t i s v a r i e db y c h a n g i n g t h e l e a k c o ndu c t a n ce g L . W i t h g r e a t e r l e a k c o ndu c t a n ce , t h e m e m b r a n e p o t e n t i a l V f o ll o w s V e f m o r ec l o s e l y a nd t h e nh i g h e r c l e a d s t o l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s m o r e un i v e r s a ll y ( a pp r o x i m a t i o n o f σ V , c a l c u l a t e d f r o m E T A ,i s c o l o r c o d e d ) . T h e r e g i o n s w h e r e h i g h e r c d ec r e a s e s / i n c r e a s e s t h e m e m b r a n e p o t e n t i a l a r e s e p a r a t e db y t h e d o tt e d li n e . B o tt o m r o w : T h e s t r u c t u r e o f t h ec o n t o u r p l o t s i nd i c a t i n g h o w fi r i n g r e g u l a r i t y ( C V , c o l o r c o d e d ) c h a n g e s w i t h c a nd P S F R f o ll o w s t h e s t r u c t u r e o f t h e m e m b r a n e p o t e n t i a l flu c t u a t i o nh e a t m a p s (t o p r o w ) . W i t hh i g h e r g L , h i g h e r c i n c r e a s e s t h e fi r i n g r e g u l a r i t y f o r a w i d e rr a n g e o f fi r i n g r a t e s a nd a w i d e rr a n g e o f c . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r fi r i n g r e g u l a r i t y . upp l e m e n t a r y F i g u r e : V a r y i n g t h e a p li t ud e o f i nh i b i t o r y s y n a p s e s . W i t h g r e a t e r a m p li t ud e o f t h e i nh i b i t o r y s y n a p s e s ( A i , E q . ( )) , t h e s t a b ili s a t i o n o f t h e m e m b r a n e p o t e n t i a li s o b s e r v e d o n l y f o r h i g h e r v a l u e s o f c (t o p r o w ) . T h i s i s r e fl ec t e d i n t h e fi r i n g r e g u l a r i t y o f t h e D T - L I F m o d e l ( b o tt o m r o w ) , a n a l ogo u s l y t o t h e Supp l e m e n t a r y F i g u r e . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s ( f o r t h e t o p r o w ) o r h i g h e r / l o w e r fi r i n g r e g u l a r i t y ( f o r t h e b o tt o m r o w ) . upp l e m e n t a r y F i g u r e : C o n t i nu o u s c h a n g e o f C V i n t h e HH - D T m o d e l. L e f t : S a m e a s F i g . C , bu t f o r c ∈ { , . , . , . , . } . D a s h e d a ndd o tt e d v e r t i c a lli n e s w i t hp o i n t s i nd i c a t ee t h ee l o w ee s t v a l u ee o f C V f o r ee a c h c . T h ee s ee a r ee t h ee
Neuronal Dy-namics (Cambridge University Press, 2019).[47] C. Teeter, R. Iyer, V. Menon, N. Gouwens, D. Feng,J. Berg, A. Szafer, N. Cain, H. Zeng, M. Hawrylycz,C. Koch, and S. Mihalas, Generalized leaky integrate-and-fire models classify multiple neuron types, Nat. Com-mun , 709 (2018).[48] J. Benda, L. Maler, and A. Longtin, Linear Versus Non-linear Signal Transmission in Neuron Models With Adap-tation Currents or Dynamic Thresholds, J. Neurophysiol , 2806 (2010).[49] A. Destexhe and D. Par´e, Impact of Network Activityon the Integrative Properties of Neocortical PyramidalNeurons In Vivo, J. Neurophysiol , 1531 (1999).[50] W. R. Softky and C. Koch, The highly irregular firing ofcortical cells is inconsistent with temporal integration ofrandom EPSPs, J. Neurosci , 334 (1993).[51] D. P. Heyman and M. J. Sobel, Stochastic Models inOperations Research: Stochastic Processes and Operat- ing Characteristics , Dover Books on Computer ScienceSeries (Dover Publications, 2004).[52] B. Lindner, Superposition of many independent spiketrains is generally not a Poisson process, Phys Rev E , 022901 (2006).[53] J. F. C. Kingman, Poisson Processes (Oxford Studies inProbability) (Clarendon Press, 1993).[54] D. J. Amit and M. V. Tsodyks, Effective neurons and at-tractor neural networks in cortical environment, Network , 121 (1992).[55] T. Toyoizumi, K. Aihara, and S.-i. Amari, Fisher Infor-mation for Spike-Based Population Decoding, Phys. Rev.Lett. , 098102 (2006).[56] L. Kostal, P. Lansky, and J.-P. Rospars, Neuronal cod-ing and spiking randomness, Eur. J. Neurosci , 2693(2007).[57] R. R. de Ruyter van Steveninck, Reproducibility andVariability in Neural Spike Trains, Science , 1805(1997).[58] S. P. Strong, R. Koberle, R. R. de Ruyter van Steveninck,and W. Bialek, Entropy and Information in Neural SpikeTrains, Phys. Rev. Lett. , 197 (1998).[59] P. H. E. Tiesinga, J. V. Jos´e, and T. J. Sejnowski, Com-parison of current-driven and conductance-driven neocor-tical model neurons with Hodgkin-Huxley voltage-gatedchannels, Phys Rev E , 8413 (2000).[60] M. J. Chacron, B. Lindner, and A. Longtin, Noise Shap-ing by Interval Correlations Increases Information Trans-fer, Phys. Rev. Lett. , 080601 (2004).[61] B. Lindner, M. J. Chacron, and A. Longtin, Integrate-and-fire neurons with threshold noise: A tractable modelof how interspike interval correlations affect neuronal sig-nal transmission, Phys Rev E , 021911 (2005).[62] F. Farkhooi, E. Muller, and M. P. Nawrot, Adaptationreduces variability of the neuronal population code, PhysRev E , 050905 (2011).[63] B. Lindner and A. Longtin, Comment on “Characteri-zation of Subthreshold Voltage Fluctuations in NeuronalMembranes,” by M. Rudolph and A. Destexhe, NeuralComput. , 1896 (2006).[64] D. Sterratt, B. Graham, A. Gillies, and D. Willshaw, Principles of Computational Modelling in Neuroscience (Cambridge University Press, 2011).[65] M. J. Chacron, A. Longtin, M. St-Hilaire, and L. Maler,Suprathreshold Stochastic Firing Dynamics with Mem-ory in P-Type Electroreceptors, Phys. Rev. Lett. , 1576(2000).[66] R. Kobayashi, Y. Tsubo, and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci. , 9 (2009).[67] W. Gerstner and R. Naud, How Good Are Neuron Mod-els?, Science , 379 (2009).[68] R. Kobayashi, S. Kurita, A. Kurth, K. Kitano,K. Mizuseki, M. Diesmann, B. J. Richmond, and S. Shi-nomoto, Reconstructing neuronal circuitry from parallelspike trains, Nat. Commun. , 4468 (2019).[69] R. Moreno-Bote and N. Parga, Membrane Potential andResponse Properties of Populations of Cortical Neuronsin the High Conductance State, Phys. Rev. Lett. ,088103 (2005).[70] R. Moreno-Bote and N. Parga, Auto- and Crosscorrel-ograms for the Spike Response of Leaky Integrate-and-Fire Neurons with Slow Synapses, Phys. Rev. Lett. , 028101 (2006).[71] R. Moreno-Bote, A. Renart, and N. Parga, Theory of In-put Spike Auto- and Cross-Correlations and Their Effecton the Response of Spiking Neurons, Neural Comput. ,1651 (2008).[72] R. Moreno-Bote and N. Parga, Response of Integrate-and-Fire Neurons to Noisy Inputs Filtered by Synapseswith Arbitrary Timescales: Firing Rate and Correlations,Neural Comput. , 1528 (2010). upp l e m e n t a r y F i g u r e : C o n s t r u c t i o n a nd v a li d a t i o n o f d y n a m i c t h r e s h o l d HH m o d e l. A - D : T h e b l u e li n e s r e p r e s e n tt h e a c t i v a t i o n ( A ) a nd i n a c t i v a t i o n ( B ) r a t e f un c t i o n s o f t h e ga t i n g v a r i a b l e h o f t h e n o n - a d a p t i n g HH m o d e l ( HH - ) . T h e o r a n g e li n e s r e p r e s e n tt h e m o d i fi e d r a t e f un c t i o n s f o r t h e HH - D T m o d e l. A s a n e ff ec t o f t h i s m o d i fi c a t i o n , t h e r ec o v e r y t i m e i n t h e HH - D T m o d e li s p r o l o n g e d ( C ) , w h il e t h e s t e a d y s t a t e a c t i v a t i o n h ∞ r e m a i n s r e l a t i v e l y un c h a n g e d ( D ) . E : C o m pu t a t i o n o f t h e t h r e s h o l d . I f n oa dd i t i o n a l p r e - s y n a p t i c s p i k e s a rr i v e a f t e r t = . m s , n o s p i k e i s o b s e r v e d ( g r ee n t r a ce ) . I f t h e p r e - s y n a p t i c s p i k e ss t o p a rr i v i n g ∆ t = . m s l a t e r , a s p i k e i s o b s e r v e d . T h e r e f o r e , w e d e fin e t h e t h r e s h o l d a s b e i n g r e a c h e d a t t = . m s . F - H : C o m p a r i s o n o f t h ec a l c u l a t e d t h r e s h o l d w i t h t h e p r ece d i n g i n t e r s p i k e i n t e r v a l ( I S I ) . W e v a li d a t e d t h a tt h e t h r e s h o l dd ec r e a s e s w i t h t i m e a f t e r a s p i k e i s fi r e db y e s t i m a t i n g t h e fi r i n g t h r e s h o l d i nd i v i du a ll y f o r e a c h s p i k e du r i n ga s i m u l a t i o n w i t h a s t a t i o n a r y i npu t , r e s u l t i n g i n a p o s t - s y n a p t i c fi r i n g r a t e o f a pp r o x i m a t e l y H z N e i t h e r t h e n o n - a d a p t i n g m o d e l ( HH - , p a n e l F ) n o r t h e m o d e l w i t h M - c u rr e n t a d a p t a t i o n ( HH - M , p a n e l G ) d i s p l a y ss i g n s o f s i g n i fi c a n t d e p e nd e n ce b e t w ee n t h e t h r e s h o l d a nd I S I . F o r t h e d y n a m i c t h r e s h o l d m o d e l ( HH - D T , p a n e l H ) w e o b s e r v e d a c l e a r d ec r e a s e i n t h e t h r e s h o l d w i t h i n c r e a s i n g du r a t i o n o f t h e p r ece d i n g I S I . T h e r e d li n e , a c t i n ga s a v i s u a l a i d , w a s o b t a i n e d b y a K NN r e g r e ss i o n . upp l e m e n t a r y F i g u r e : V a r y i n g t h e m e m b r a n e t i m e c o n s t a n t . T o p r o w : T h e m e m b r a n e t i m ec o n s t a n t i s v a r i e db y c h a n g i n g t h e l e a k c o ndu c t a n ce g L . W i t h g r e a t e r l e a k c o ndu c t a n ce , t h e m e m b r a n e p o t e n t i a l V f o ll o w s V e f m o r ec l o s e l y a nd t h e nh i g h e r c l e a d s t o l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s m o r e un i v e r s a ll y ( a pp r o x i m a t i o n o f σ V , c a l c u l a t e d f r o m E T A ,i s c o l o r c o d e d ) . T h e r e g i o n s w h e r e h i g h e r c d ec r e a s e s / i n c r e a s e s t h e m e m b r a n e p o t e n t i a l a r e s e p a r a t e db y t h e d o tt e d li n e . B o tt o m r o w : T h e s t r u c t u r e o f t h ec o n t o u r p l o t s i nd i c a t i n g h o w fi r i n g r e g u l a r i t y ( C V , c o l o r c o d e d ) c h a n g e s w i t h c a nd P S F R f o ll o w s t h e s t r u c t u r e o f t h e m e m b r a n e p o t e n t i a l flu c t u a t i o nh e a t m a p s (t o p r o w ) . W i t hh i g h e r g L , h i g h e r c i n c r e a s e s t h e fi r i n g r e g u l a r i t y f o r a w i d e rr a n g e o f fi r i n g r a t e s a nd a w i d e rr a n g e o f c . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r fi r i n g r e g u l a r i t y . upp l e m e n t a r y F i g u r e : V a r y i n g t h e a p li t ud e o f i nh i b i t o r y s y n a p s e s . W i t h g r e a t e r a m p li t ud e o f t h e i nh i b i t o r y s y n a p s e s ( A i , E q . ( )) , t h e s t a b ili s a t i o n o f t h e m e m b r a n e p o t e n t i a li s o b s e r v e d o n l y f o r h i g h e r v a l u e s o f c (t o p r o w ) . T h i s i s r e fl ec t e d i n t h e fi r i n g r e g u l a r i t y o f t h e D T - L I F m o d e l ( b o tt o m r o w ) , a n a l ogo u s l y t o t h e Supp l e m e n t a r y F i g u r e . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s ( f o r t h e t o p r o w ) o r h i g h e r / l o w e r fi r i n g r e g u l a r i t y ( f o r t h e b o tt o m r o w ) . upp l e m e n t a r y F i g u r e : C o n t i nu o u s c h a n g e o f C V i n t h e HH - D T m o d e l. L e f t : S a m e a s F i g . C , bu t f o r c ∈ { , . , . , . , . } . D a s h e d a ndd o tt e d v e r t i c a lli n e s w i t hp o i n t s i nd i c a t ee t h ee l o w ee s t v a l u ee o f C V f o r ee a c h c . T h ee s ee a r ee t h ee n s h o w n i n t h ee
Neuronal Dy-namics (Cambridge University Press, 2019).[47] C. Teeter, R. Iyer, V. Menon, N. Gouwens, D. Feng,J. Berg, A. Szafer, N. Cain, H. Zeng, M. Hawrylycz,C. Koch, and S. Mihalas, Generalized leaky integrate-and-fire models classify multiple neuron types, Nat. Com-mun , 709 (2018).[48] J. Benda, L. Maler, and A. Longtin, Linear Versus Non-linear Signal Transmission in Neuron Models With Adap-tation Currents or Dynamic Thresholds, J. Neurophysiol , 2806 (2010).[49] A. Destexhe and D. Par´e, Impact of Network Activityon the Integrative Properties of Neocortical PyramidalNeurons In Vivo, J. Neurophysiol , 1531 (1999).[50] W. R. Softky and C. Koch, The highly irregular firing ofcortical cells is inconsistent with temporal integration ofrandom EPSPs, J. Neurosci , 334 (1993).[51] D. P. Heyman and M. J. Sobel, Stochastic Models inOperations Research: Stochastic Processes and Operat- ing Characteristics , Dover Books on Computer ScienceSeries (Dover Publications, 2004).[52] B. Lindner, Superposition of many independent spiketrains is generally not a Poisson process, Phys Rev E , 022901 (2006).[53] J. F. C. Kingman, Poisson Processes (Oxford Studies inProbability) (Clarendon Press, 1993).[54] D. J. Amit and M. V. Tsodyks, Effective neurons and at-tractor neural networks in cortical environment, Network , 121 (1992).[55] T. Toyoizumi, K. Aihara, and S.-i. Amari, Fisher Infor-mation for Spike-Based Population Decoding, Phys. Rev.Lett. , 098102 (2006).[56] L. Kostal, P. Lansky, and J.-P. Rospars, Neuronal cod-ing and spiking randomness, Eur. J. Neurosci , 2693(2007).[57] R. R. de Ruyter van Steveninck, Reproducibility andVariability in Neural Spike Trains, Science , 1805(1997).[58] S. P. Strong, R. Koberle, R. R. de Ruyter van Steveninck,and W. Bialek, Entropy and Information in Neural SpikeTrains, Phys. Rev. Lett. , 197 (1998).[59] P. H. E. Tiesinga, J. V. Jos´e, and T. J. 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T h e o r a n g e li n e s r e p r e s e n tt h e m o d i fi e d r a t e f un c t i o n s f o r t h e HH - D T m o d e l. A s a n e ff ec t o f t h i s m o d i fi c a t i o n , t h e r ec o v e r y t i m e i n t h e HH - D T m o d e li s p r o l o n g e d ( C ) , w h il e t h e s t e a d y s t a t e a c t i v a t i o n h ∞ r e m a i n s r e l a t i v e l y un c h a n g e d ( D ) . E : C o m pu t a t i o n o f t h e t h r e s h o l d . I f n oa dd i t i o n a l p r e - s y n a p t i c s p i k e s a rr i v e a f t e r t = . m s , n o s p i k e i s o b s e r v e d ( g r ee n t r a ce ) . I f t h e p r e - s y n a p t i c s p i k e ss t o p a rr i v i n g ∆ t = . m s l a t e r , a s p i k e i s o b s e r v e d . T h e r e f o r e , w e d e fin e t h e t h r e s h o l d a s b e i n g r e a c h e d a t t = . m s . F - H : C o m p a r i s o n o f t h ec a l c u l a t e d t h r e s h o l d w i t h t h e p r ece d i n g i n t e r s p i k e i n t e r v a l ( I S I ) . W e v a li d a t e d t h a tt h e t h r e s h o l dd ec r e a s e s w i t h t i m e a f t e r a s p i k e i s fi r e db y e s t i m a t i n g t h e fi r i n g t h r e s h o l d i nd i v i du a ll y f o r e a c h s p i k e du r i n ga s i m u l a t i o n w i t h a s t a t i o n a r y i npu t , r e s u l t i n g i n a p o s t - s y n a p t i c fi r i n g r a t e o f a pp r o x i m a t e l y H z N e i t h e r t h e n o n - a d a p t i n g m o d e l ( HH - , p a n e l F ) n o r t h e m o d e l w i t h M - c u rr e n t a d a p t a t i o n ( HH - M , p a n e l G ) d i s p l a y ss i g n s o f s i g n i fi c a n t d e p e nd e n ce b e t w ee n t h e t h r e s h o l d a nd I S I . F o r t h e d y n a m i c t h r e s h o l d m o d e l ( HH - D T , p a n e l H ) w e o b s e r v e d a c l e a r d ec r e a s e i n t h e t h r e s h o l d w i t h i n c r e a s i n g du r a t i o n o f t h e p r ece d i n g I S I . T h e r e d li n e , a c t i n ga s a v i s u a l a i d , w a s o b t a i n e d b y a K NN r e g r e ss i o n . upp l e m e n t a r y F i g u r e : V a r y i n g t h e m e m b r a n e t i m e c o n s t a n t . T o p r o w : T h e m e m b r a n e t i m ec o n s t a n t i s v a r i e db y c h a n g i n g t h e l e a k c o ndu c t a n ce g L . W i t h g r e a t e r l e a k c o ndu c t a n ce , t h e m e m b r a n e p o t e n t i a l V f o ll o w s V e f m o r ec l o s e l y a nd t h e nh i g h e r c l e a d s t o l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s m o r e un i v e r s a ll y ( a pp r o x i m a t i o n o f σ V , c a l c u l a t e d f r o m E T A ,i s c o l o r c o d e d ) . T h e r e g i o n s w h e r e h i g h e r c d ec r e a s e s / i n c r e a s e s t h e m e m b r a n e p o t e n t i a l a r e s e p a r a t e db y t h e d o tt e d li n e . B o tt o m r o w : T h e s t r u c t u r e o f t h ec o n t o u r p l o t s i nd i c a t i n g h o w fi r i n g r e g u l a r i t y ( C V , c o l o r c o d e d ) c h a n g e s w i t h c a nd P S F R f o ll o w s t h e s t r u c t u r e o f t h e m e m b r a n e p o t e n t i a l flu c t u a t i o nh e a t m a p s (t o p r o w ) . W i t hh i g h e r g L , h i g h e r c i n c r e a s e s t h e fi r i n g r e g u l a r i t y f o r a w i d e rr a n g e o f fi r i n g r a t e s a nd a w i d e rr a n g e o f c . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r fi r i n g r e g u l a r i t y . upp l e m e n t a r y F i g u r e : V a r y i n g t h e a p li t ud e o f i nh i b i t o r y s y n a p s e s . W i t h g r e a t e r a m p li t ud e o f t h e i nh i b i t o r y s y n a p s e s ( A i , E q . ( )) , t h e s t a b ili s a t i o n o f t h e m e m b r a n e p o t e n t i a li s o b s e r v e d o n l y f o r h i g h e r v a l u e s o f c (t o p r o w ) . T h i s i s r e fl ec t e d i n t h e fi r i n g r e g u l a r i t y o f t h e D T - L I F m o d e l ( b o tt o m r o w ) , a n a l ogo u s l y t o t h e Supp l e m e n t a r y F i g u r e . T h e d o tt e d li n e a pp r o x i m a t e s t h e s e p a r a t i o nb e t w ee n t h e r e g i o n s w h e r e h i g h e r c l e a d s t o h i g h e r / l o w e r m e m b r a n e p o t e n t i a l flu c t u a t i o n s ( f o r t h e t o p r o w ) o r h i g h e r / l o w e r fi r i n g r e g u l a r i t y ( f o r t h e b o tt o m r o w ) . upp l e m e n t a r y F i g u r e : C o n t i nu o u s c h a n g e o f C V i n t h e HH - D T m o d e l. L e f t : S a m e a s F i g . C , bu t f o r c ∈ { , . , . , . , . } . D a s h e d a ndd o tt e d v e r t i c a lli n e s w i t hp o i n t s i nd i c a t ee t h ee l o w ee s t v a l u ee o f C V f o r ee a c h c . T h ee s ee a r ee t h ee n s h o w n i n t h ee r i g h t p a n ee