Multicoding in neural information transfer suggested by mathematical analysis of the frequency-dependent synaptic plasticity in vivo
Katsuhiko Hata, Osamu Araki, Osamu Yokoi, Tatsumi Kusakabe, Yoshio Yamamoto, Susumu Ito, Tetsuro Nikuni
aa r X i v : . [ q - b i o . N C ] J a n M ULTIC ODING IN NEUR AL INFOR MATION TR ANSFERSUGGESTED B Y MATHEMATICAL ANALYSIS OF THEFR EQUENCY - DEPENDENT SYNAPTIC PLASTICITY IN VIVO
A P
REPRINT
Katsuhiko Hata ∗
1, 3, 5 , Osamu Araki , Osamu Yokoi , Tatsumi Kusakabe , Yoshio Yamamoto , Susumu Ito , andTetsuro Nikuni Department of Sports and Medical Science, Kokushikan University, Tokyo, Japan Department of Applied Physics, Faculty of Science Division I, Tokyo University of Science, Tokyo, Japan Department of Neuroscience, Research Center for Mathematical Medicine, Tokyo, Japan Laboratory of Veterinary Biochemistry and Cell Biology, Faculty of Agriculture, Iwate University, Morioka, Japan Department of Physics, Faculty of Science Division I, Tokyo University of Science, Tokyo, JapanJanuary 14, 2020 A BSTRACT
Two elements of neural information processing have primarily been proposed: firing rate and spiketiming of neurons. In the case of synaptic plasticity, although spike-timing-dependent plasticity(STDP) depending on presynaptic and postsynaptic spike times had been considered the most com-mon rule, recent studies have shown the inhibitory nature of the brain in vivo for precise spiketiming, which is key to the STDP. Thus, the importance of the firing frequency in synaptic plasticityin vivo has been recognized again. However, little is understood about how the frequency-dependentsynaptic plasticity (FDP) is regulated in vivo. Here, we focused on the presynaptic input pattern, theintracellular calcium decay time constants, and the background synaptic activity, which vary depend-ing on neuron types and the anatomical and physiological environment in the brain. By analyzing acalcium-based model, we found that the synaptic weight differs depending on these factors charac-teristic in vivo, even if neurons receive the same input rate. This finding suggests the involvementof multifaceted factors other than input frequency in FDP and even neural coding in vivo. ∗ [email protected] PREPRINT - J
ANUARY
14, 2020
Synaptic plasticity in neural networks is a substrate of learning and memory, which includes both positive and nega-tive components, i.e., both long-lasting enhancements and declines in the weight of synaptic transmission (long-termpotentiation (LTP) and long-term depression (LTD)) [1]. Many experimental studies have suggested two plausiblemechanisms for the induction of the synaptic plasticity [2, 3]. The first is the frequency of spike trains, which hasbeen studied in association with the Bienenstock, Cooper, and Munro (BCM) rule in classical research conductedapproximately half a century ago [3, 4, 5, 6]. LTP is induced by high-frequency firing in presynaptic neurons, whichproduces large increases in postsynaptic calcium concentration [5, 6, 7, 8]. The low-frequency firing causes a modestincrease in the calcium level, and thereby induces LTD [9, 10, 11]. The second is the precise timing of presynapticand postsynaptic firing, which has been investigated as spike-time-dependent plasticity (STDP) in numerous experi-mental and theoretical studies from approximately 20 years ago [3, 12, 13]. LTP is induced by the presynaptic actionpotentials preceding postsynaptic spikes by no more than tens of milliseconds, whereas presynaptic firing that followspostsynaptic spikes produces LTD [13, 14, 15, 16, 17, 18]. The idea that STDP plays a central role in synaptic plas-ticity had been becoming mainstream. Recent studies have reported, however, that in some cases, the environmentin vivo may not be suitable for precise spike timing, which is key to the STDP. Pre- and post-synaptic neurons in theprimary visual cortex and extrastriate cortex of awaking animals fire so irregularly that the timing of presynaptic andpostsynaptic firing varies [19, 20, 21]. Neurons and synapses in the cerebral cortex of rats receive a lot of backgroundneuronal activity that is generated internally, which provides strong constraints on spike timing [22, 23, 24]. In theseenvironments, the firing rate, rather than the spike timing, is likely to be important for the synaptic plasticity and neuralcoding. For example, it has been demonstrated experimentally that the cerebral cortex in which there is a high level ofinternal noise uses a rate code [24], and it has been shown mathematically that synaptic changes are induced by varia-tion of firing rate without any timing constraints [19]. Moreover, firing variability, as well as the statistical propertiesof the spike frequency, may be important for real-time information processing [25]. Based on these reports, the role offiring frequency in various aspects of neural information processing has again come into the limelight. Furthermore,in vivo characteristic factors such as the variation of the firing pattern, the difference of intracellular parameters, andinternal noise have also been suggested to be important for synaptic plasticity and neural coding. However, how thesefactors are involved in the synaptic plasticity is poorly understood. In order to clarify this problem, we examinedthe role of the presynaptic input pattern, the intracellular calcium decay time constants, and the background synapticactivity in frequency-dependent synaptic plasticity (FDP) by analyzing a calcium-based model, which is one of themost compatible models with experimental results [12].Currently, it is widely accepted that the calcium concentration in the postsynapse determines whether LTP or LTD isinduced [26, 27, 28, 29]. A moderate elevation of intracellular calcium correlates with induction of LTD, whereas alarger increase correlates with LTP [27, 28]. Only if glutamate is released by presynaptic activity and if the postsynap-tic membrane is depolarized sufficiently, calcium ions enter the cell through channels controlled by NMDA receptors[12]. The depolarization of the postsynaptic membrane potential is due not only to excitatory postsynaptic potentials(EPSPs) generated by binding glutamate to the AMPA receptors but also to many kinds of background synaptic activ-ities [30, 31, 32, 33]. These experimental events were formulated by Shouval et al. [34] as a calcium-based model,which has been used in numerous studies.In the present study, we investigated the FDP in vivo analytically and numerically using the calcium-based model.First, in order to investigate the FDP in neurons with in vivo-specific firing pattern, we used three types of firing,which are widely observed in the brain, that is, constant-inter-spike intervals (ISI) inputs, Poisson inputs, and gammainputs. Next, the calcium decay time constant of in vivo neurons varies from cell to cell. Previous reports suggestedthat pyramidal neurons in superficial layers possess faster calcium dynamics than those in deep layers. Here, τ ca ≈ ms in layer II to IV neurons, whereas τ ca ≈ ms in layer V to VI neurons [35, 36]. In order to study the associationof the calcium decay time constant with the FDP, we examined two kinds of neurons with time constants of 40 ms and80 ms. Finally, neurons in vivo are constantly exposed to background synaptic activity [30, 33]. The frequency andmagnitude of this activity vary depending on the location of the synapse and the level of neuronal activity [30, 33].We therefore examined the correlation between the amplitude of background activity and the FDP. The findings in thepresent study may contribute to a detailed understanding of synaptic plasticity in in vivo brain.1 PREPRINT - J
ANUARY
14, 2020
In Figure 1, we plot the analytical solution of Ca in Eq. (39) as a function of the input frequency f . We also plot thesimulation results obtained by solving Eqs. (5)-(14) numerically as a function of time and taking the time average of Ca for each frequency. The analytical solution for the long-term behavior of calcium level agrees very well with thenumerical simulation results. We adopted τ Ca = 80 ms for a long calcium decay time constant and τ Ca = 40 ms asa short calcium decay time constant. The calcium concentration as a function of input frequency increases slower for τ Ca = 40 ms than for τ Ca = 80 ms. Equation (39) indicates that the calcium concentration at an arbitrary stimulationrate increases linearly for the calcium time constant τ Ca .Figure 1: Presynaptic firing rate-induced elevation of intracellular calcium concentration in two types of neuronswith different time constants of calcium decay. The analytic solution is indicated by solid lines, while the resultsof numerical calculation are indicated by dotted lines. The calcium level increases more slowly in neurons with theshort calcium decay time constant ( ms) than in neurons with the long decay time constant ( ms). This is alsounderstood from Eq. (39). Error bars indicate the standard error of the mean (SEM). Figure 2 shows the curve obtained by performing the integration in Eq. (48). We also plot the results obtained by thenumerical simulation, which agree qualitatively with the analytical results. These results suggest that the LTD/LTPthreshold shifts to a lower frequency as the calcium time constant increases. Here, the LTD/LTP threshold is definedas the frequency at which the synaptic weight first returns to 1 after falling below 1 when the input frequency isincreased from 0 Hz. This tendency can also be understood from Eq. (39) as follows. Equation (39) is written as Ca ( f ) = τ Ca · F ( f ) , where F ( f ) is a monotonically increasing function of f , so that f can be formally expressed as f = F − ( Ca/τ Ca ) . Equations (7) and (46) indicate that when the synaptic strength is at the LTD/LTP threshold, thepostsynaptic calcium level has a fixed value: Ca = 1 β log e βα − . e βα . . (1)Substituting the numerical values of the parameters in Eq. (7) into Eq. (1), we obtained Ca = 0 . µ M. Thus, thestimulation frequency when the synaptic weight reaches the LTD/LTP threshold is a monotonically increasing functionof /τ Ca . In several experimental studies on synaptic plasticity, the paradigms for inducing synaptic plasticity have consistedof constant-frequency stimulation trains, such as paired pulses or a tetanic stimulus. Neurons in vivo, however, areunlikely to experience such simple inputs. Rather, these neurons receive more complex input patterns in which ISIs arehighly irregular [37]. The most representative stimulation patterns that are not constant-frequency stimulation trainsare the Poisson process and the gamma process. In fact, spike sequences similar to these processes are sometimes2
PREPRINT - J
ANUARY
14, 2020Figure 2: Synaptic strength in two types of neurons that have different calcium decay time constants as a functionof the constant presynaptic stimulation frequency. The x axis indicates the input frequency, and the y axis representsnormalized synaptic weights that are obtained after several hundreds of presynaptic spikes. The analytic solutions areindicated by solid lines, whereas the solutions provided by numerical calculation are indicated by dotted lines. Errorbars indicate the SEM.observed in neurons of brain [12, 38, 39, 40, 41]. In this section, we discuss the results for the FDP of neurons withPoisson-distributed spike trains.First, the calcium concentration at the postsynapse receiving Poisson input was calculated numerically, and is plottedin Fig. 3, in which the calcium concentration with constant-ISI input is also plotted for comparison. In the samemanner, we examined two kinds of neurons with calcium time constants of 40 ms and 80 ms. The intracellularcalcium concentration, regardless of the stimulation pattern, increases more gradually in the case of τ Ca = 40 ms thanin the case of τ Ca = 80 ms. In addition, the calcium level with Poisson input increases more slowly than that withconstant-ISI input, which is independent of the calcium time constant (Figs. 3B and 3C).Next, we examined the strength of a synapse receiving Poisson input. In Fig. 4, we define the LTD phase or LTP phaseas the range of frequency indicating LTD or LTP. When the calcium time constant is ms, interestingly, Poissoninput makes the LTD phase disappear and the LTP phase is observed at any input frequency, whereas in the case ofconstant-ISI stimulation, the LTD phase still exists at roughly between 3 Hz and 9 Hz (see Fig. 4B, left panel). Whenthe calcium time constant is 40 ms, unlike in the case of τ Ca = 80 ms, changing the stimulus pattern from constant-ISI input to Poisson input shifted the LTD/LTP threshold to the right (see Fig. 4B, right panel). Since the firing rateobserved in the brain is found to be at most approximately 112 Hz, we need only consider synaptic plasticity within100 Hz [42]. This consideration leads to the conclusion that Poisson input to a neuron with τ Ca = 40 ms expandsthe LTD phase and narrows the LTP phase. These results can be well reproduced by approximate analytical solutions(Eqs. (41) and (49)).Analytical solutions for the calcium concentration with Poisson input (Eq. (41)) are plotted in Fig. 3A. As shown inFig. 3C, the solutions agree well with the numerical results and indicate that Poisson stimulation gently increases thecalcium concentration, as compared to constant-ISI input. This property does not depend on the calcium decay timeconstant (Fig. 5).Next, we obtained an approximate expression for the relation between the synaptic weight and the average stimulationrate. By assuming that the synaptic weight W ( t ) converges to a stationary solution in the long-time scale (Eq. (46)), weobtain Eq. (49). Figures 4A and 4C show that the analytical expression agrees well with the results of the numericalsimulation. Regardless of the τ Ca value, the synaptic weight varies slowly by changing the stimulus pattern fromconstant-ISI input to Poisson input. This change in the stimulation pattern moves the LTD/LTP threshold to the leftand narrows the LTD phase decrease for τ Ca = 80 ms (Fig. 4A, left), whereas it has the opposite effect for τ Ca = 40 ms (Fig. 4A, right).Thus, the numerical and analytical studies indicate that the postsynaptic calcium concentration and synaptic strengthreceiving Poisson input behave differently from those receiving constant-ISI stimulation. At the same frequency, when τ Ca = 80 ms, a synapse receiving Poisson input is more likely to be LTP than a synapse receiving constant-ISI input,and when τ Ca = 40 ms, a synapse receiving Poisson input is more likely to be LTD. These findings suggest thatthe difference in input patterns (constant-ISI or Poisson input) and calcium decay time constant affects the output ofFDP, i.e., LTD or LTP. In addition, this tendency to become LTP or LTD by changing the input pattern depends on thepostsynaptic calcium decay time constant. 3 PREPRINT - J
ANUARY
14, 2020Figure 3: Relation between the postsynaptic calcium concentration and the average frequency of presynaptic constant-ISI (or Poisson) input. As in Fig. 1, two types of neurons with different calcium time constants were examined. Theanalytic solution is shown in A and C, whereas the results of numerical simulation are shown in B and C. The blue(or red) lines indicate the calcium concentration of the postsynapse with constant-ISI (or Poisson) input. In the case ofPoisson input, the increase in calcium concentration with respect to the average frequency is slower than in the case ofconstant-ISI input. This result is independent of the calcium time constant. Error bars indicate the SEM.4
PREPRINT - J
ANUARY
14, 2020Figure 4: Synaptic strength for two types of neurons ( τ Ca = 80 ms and τ Ca = 40 ms) as a function of theaverage rate of presynaptic stimulation. The x axis represents the input frequency, and the y axis represents normalizedsynaptic weights that are obtained after several hundreds of presynaptic spikes. The analytic solution is shown in Aand C, whereas the results of numerical simulation are shown in B and C. Error bars indicate the SEM. The blue (orred) lines indicate the synaptic weights with constant-ISI (or Poisson) input. The synaptic weight with the Poissoninput changes slowly compared to that with the constant-ISI input. As shown by the numerical simulation results, inneurons with τ Ca = 80 ms, the Poisson input makes the LTD phase disappear, and only the LTP phase remains (B,left panel). On the other hand, in neurons with τ Ca = 40 ms, the LTD/LTP threshold moves to the right, and the LTDphase increases (B, right panel). These results are also qualitatively illustrated by analytical solutions (A).5 PREPRINT - J
ANUARY
14, 2020Figure 5: Two-dimensional density plot of post-synaptic calcium concentration as a function of f and τ Ca . Weillustrate the calcium concentration with the constant-ISI input expressed in Eq. (39) (A) and with the Poisson inputexpressed in Eq. (41) (B). (C) Density plot of h Ca c ( f, τ Ca ) i − h Ca poi ( f, τ Ca ) i . The horizontal lines on each figuresuggest corresponding values at τ Ca = 40 ms and τ Ca = 80 . We studied the postsynaptic calcium concentration and synaptic load of neurons receiving gamma process inputs,which is one of the firing patterns observed in brain [41, 43]. Since the analytic solutions are qualitatively consistentwith the simulation results so far presented in the present paper, we discuss the plasticity of synapses receiving gammaprocess input by only the analytic solutions. The postsynaptic calcium concentration of neurons that receive gammaprocess input is expressed by Eq. (45), where α is a shape parameter. The synaptic weight of the neurons receivinggamma process input is approximately expressed by Eq. (50) as a function of average input frequency.This result for the calcium concentration is illustrated in Fig. 6A. As the shape parameter increases, the slope ofthe calcium concentration increases. The results for the synaptic weight are shown in Fig. 6B. When neurons with τ Ca = 80 ms are stimulated by gamma process input, as the shape parameter α increases, the LTD/LTP thresholdshifts to a higher frequency and the minimum value of the synaptic weight becomes smaller (Fig. 6B, left). When τ Ca = 40 ms, the LTD/LTP threshold shifts to a lower frequency as the shape parameter α increases; on the otherhand, the minimum value of the synaptic weight is approximately the same from α = 1 to α = 5 (Fig. 6B, right).In summary, the postsynaptic calcium level with gamma process input increases slower than that with constant-ISIinput, but increases faster than that with Poisson input. As the shape parameter increases, the increase in the calciumconcentration becomes faster. The tendency to induce LTP or LTD by gamma process input depends on the shapeparameter. These results suggest that the difference in input pattern as well as the shape parameter in gamma processinput affects the synaptic weight. The postsynaptic terminals in neurons in vivo display intense background activity, which is characterized by fluctua-tions in the postsynaptic membrane potential. This background activity has at least three components: dendritic actionpotential, BPAPs, and voltage noise [33, 44]. The voltage noise includes the stochastic properties of ion channels, therandom release of neurotransmitter, and thermal noise. The distance from the soma or the differences in the corticallayer, in which neurons are located, affects the frequency and size of the amplitude of the background synaptic activity[30, 31, 32].In order to examine the FDP under various background synaptic activities, we first analytically and numerically calcu-lated the dependence of the postsynaptic calcium concentration on the constant-ISI input under various frequencies ofbackground Poisson input. The fluctuation of the membrane potential due to background synaptic activity is denotedby V bg in Eq. (14). Since V bg increases in proportion to the average frequency of the background synaptic activity f bg , H ( V ) in Eq. (34) is approximately expressed as a bivariate quadratic function of f and f bg . Thus, the postsynapticcalcium concentration is given as a function of f and f bg as follows:6 PREPRINT - J
ANUARY
14, 2020Figure 6: Change in the postsynaptic calcium concentration and the weight in the synapse with gamma processinput. We show two types of neurons with different time constants of calcium decay, τ Ca = 80 ms and ms. In eachgraph, the black, orange, light blue, blue green, yellow, and blue lines indicate constant-ISI input, shape parameter α = 1 , α = 2 , α = 3 , α = 4 , and α = 5 , respectively. (A) Relationship between the postsynaptic intracellularcalcium concentration and input frequency f in neurons stimulated with gamma process input. A graph of constant-ISI stimulation is shown as a control (black lines). The trace of the shape parameter α = 1 matches the graph of thePoisson input. As the value of the shape parameter increases, the calcium level increases is faster. (B) Approximaterelationship between synaptic weight and mean input frequency in neurons with constant-ISI and gamma processinputs. The LTD/LTP threshold moves to a higher frequency in the case of τ Ca = 80 ms and the moves lower in thecase of τ Ca = 40 ms as the value of the shape parameter becomes large. h Ca c ( f, f bg ) i = τ Ca f ( ζ + ζ f + ζ f bg + ζ f + ζ f f bg + ζ f bg ) × X j = f,s I j τ j (cid:20) − exp (cid:18) − τ j · f (cid:19)(cid:21) , (2)where ζ = 1 . × − , ζ = 2 . × − , ζ = 6 . × − , ζ = 3 . × − , ζ = 1 . × − , and ζ = 1 . × − . Figure 7A plots Eq. (2) using τ Ca = 80 ms or τ Ca = 40 ms. In both cases, the higher theaverage frequency of the background Poisson input is, the faster the rate of increase in the calcium concentration withsynaptic input frequency becomes. As shown in Fig. 7B, qualitatively consistent results were obtained by numericalsimulations.We next analytically and numerically calculated the relation between the synaptic weight and the input frequencyunder various background input rates. The approximate analytic solution is obtained as follows:7 PREPRINT - J
ANUARY
14, 2020Figure 7: Postsynaptic calcium concentration in two types of neurons ( τ Ca = 80 ms and τ Ca = 40 ms) as a functionof the frequency of presynaptic input and of the background input. The ISI of the presynaptic input is constant. Thebackground Poisson input with a frequency in the range of 1 to 5 Hz was applied. The analytic solution is shown in A,whereas the results of numerical simulation are shown in B. Error bars indicate the SEM. h W c ( f, f bg ) i = Z ∞ dx Z dǫ δ (1 − x )Ω( Ca c ( f, f bg , x, ǫ | r Ca ; c , r j ; c )) . (3)Here, Ca c ( f, f bg , x, ǫ | r Ca ; c , r j ; c ) is defined by Eq. (2) in Eq. (47). More explicitly, Ca c ( f, f bg , x, ǫ | r Ca ; c , r j ; c ) isgiven by Ca c ( f, f bg , x, ǫ | r Ca ; c , r j ; c ) =( ζ + ζ f + ζ f bg + ζ f + ζ f f bg + ζ f bg ) X j = f,s I j τ j (cid:26) exp( − ǫτ j f ) − exp( − ǫτ Ca f ) + exp( − ǫτ Ca f ) (cid:20) exp( − xτ j f ) − exp( − xτ Ca f ) (cid:21) + exp( − x + ǫτ Ca f ) (cid:18) r j − r Ca − r Ca (cid:19)(cid:27) . (4)The analytical solution (3) is plotted in Fig. 8A, and the corresponding numerical solution is shown in Fig. 8B.Although two types of neurons with different calcium time constants were examined, the influence on the synapticstrengths by the increase of the background input level is qualitatively common to both types of neurons. In otherwords, the increase in the background input rate moves the LTD/LTP threshold to the left, decreases the LTD phase,and broadens the LTP phase.Thus, upregulation of background synaptic activities leads to the enhancement of synaptic efficacy through the accel-eration of the increasing rate of postsynaptic calcium concentration. These results suggest that the FDP output (LTPor LTD) varies depending on the magnitude of the applied background noise, even if the input frequency is the same.8 PREPRINT - J
ANUARY
14, 2020Figure 8: Synaptic strength as a function of the frequency of presynaptic constant-ISI input and of the backgroundPoisson input, under the background Poisson input with a frequency in the range of 1 to 5 Hz. Two types of neurons( τ Ca = 80 ms and τ Ca = 40 ms) were examined. The analytic solution and the results of numerical calculation areshown in A and B, respectively. Error bars indicate the SEM. Here, we summarize the findings of the present study: (1) We obtained approximately analytical solutions of theintracellular calcium concentration and the synaptic weight as a function of the frequency of three kinds of input:constant-ISI, Poisson, and gamma process input. The latter two input patterns are often observed in vivo. (2) In allthree input patterns, LTP occurs at a lower frequency as the calcium decay time constant increases. We used msas the longer calcium decay time constant ( = τ Ca ) and ms as the shorter calcium decay time constant. (3) Theintracellular calcium level increases more slowly in neurons with Poisson input than in neurons with constant-ISIinput. At the same stimulation frequency, a synapse with a long calcium time constant tends to be strengthened (LTP)by changing the stimulation pattern from constant-ISI input to Poisson input, while a synapse with a short calciumtime constant weakened (LTD). (4) The calcium level with gamma process input increases faster than that with Poissoninput but slower than that with constant-ISI input. Moreover, calcium level with gamma process also increases fasteras the shape parameter grows. As the shape parameter increases, the LTD/LTP threshold moves to a higher frequencyin τ Ca = 80 ms neurons but moves to a lower frequency in τ Ca = 40 ms neurons. The minimum value of thesynaptic weight is smaller in τ Ca = 80 ms neurons but is approximately constant in τ Ca = 40 -ms neurons as theshape parameter increases. (5) The increase of background synaptic activities induces the acceleration of the increaserate of the calcium level and the enhancement of synaptic weight.These findings indicate that the synaptic weight by FDP depends not only on input frequency but also on input pattern,shape parameter in gamma process input, calcium decay time constant, and background synaptic activity, which havebeen suggested to vary in vivo depending on the location, the internal state, and the external environment of the neuron[33, 35, 36, 41, 43, 44]. In the subsequent subsections, we discuss the involvement of these factors in synaptic plasticityand neural coding.For a long time, there has been a debate on the nature of neural coding, which is primarily founded on the generation,propagation, and processing of spikes [45, 46, 47]. The classical view of neural coding emphasizes the informationcarried by the rate at which neurons produce action potentials, whereas spike variability and background activity9 PREPRINT - J
ANUARY
14, 2020were ignored or treated as noise [25, 48, 49]. In experimental and theoretical studies of recent decades, arguing theimportance of the spike timing rather than the firing rate in neural coding, the spike variability and background activityare also considered as noise activities [12, 24]. However, the results of recent electrophysiological experiments onwaking animals suggest that they are too large to be ignored for precise spike timing [24, 44], leading to a renewedawareness of the importance of the rate coding, which is less affected by individual spike variability and backgroundnoise [19, 20]. Moreover, recent studies reveal the need for several simultaneous codes (multi-coding), including spikevariability and fluctuation of membrane potential, as sources [45, 50, 51, 52]. Hence, the multi-coding hypothesis forthe neural coding problem may be supported by the results of the present study, suggesting that not only firing rate butalso firing variability, the internal parameters of neurons, and the magnitude of background synaptic activity could beimportant for neural coding and synaptic plasticity [25, 45].We found that the calcium decay time constant determines the plasticity outcome. In neurons with a long time constant,LTP is induced even by a small presynaptic rate (about 9 Hz), because the calcium concentration via the NMDAreceptors increases faster in these neurons than in neurons with a short time constant (Figs. 1 and 2). In neurons witha short time constant, LTP is not induced until the stimulation frequency is large (over about 50 Hz). This differencedue to calcium dynamics is more pronounced when the stimulation pattern is set to Poisson or gamma process input(Figs. 4 and 6).The calcium decay time constant is closely related to the function of sodium-calcium exchangers (NCXs) [53]. Sodium-calcium exchangers, which are expressed highly in dendrites and dendritic spines in a variety of brain regions [54],are controlled in activity by various intracellular and extracellular signaling molecules [55] and are widely involvedin many neural events from developmental processes to cognitive abilities [56, 57]. Thus, the calcium decay timeconstant differs depending on anatomical and physiological characteristics. Indeed, previous reports suggest that thecalcium decay time constant varies with the depth of the cerebral cortex and that nitric oxide stimulates the increaseof the calcium decay time constant in a cGMP-dependent manner [36, 55, 58]. Our findings and those of previousstudies suggest that, even with the same frequency, the synaptic plasticity induced thereby depends on the anatomicaland physiological factors and that this difference becomes more prominent when the stimulation pattern is irregular.Previous studies have demonstrated that applying an appropriate level of noise to the postsynapse results in the en-hancement of the neural sensitivity and the improvement of signal detection in the central nervous system [59, 60].Consistent with these findings, our research indicates that increased synaptic noise is more likely to induce LTP, re-gardless of the calcium time constant. Recently, the dendritic action potential has been considered as one of the maincomponents of synaptic noise. In the record of the dendritic membrane potential of freely behaving rats, dendritespikes accompanied by large subthreshold membrane potential fluctuations occur with high rates greater than theBPAP evoked in the soma [44]. In addition, it has been shown in hippocampal synapses that even a single presynapticburst induces LTP, provided dendritic action potentials are generated [61]. These findings and our results indicate thatinputs from other than the presynapse, such as background synaptic activity, including the BPAP and the dendriticaction potential, are largely involved in synaptic plasticity, especially the generation of LTP. We cannot, however, con-clude from our results that even a single presynaptic input induces LTP. It is necessary to conduct research in whichsingle-burst-induced LTP is substantiated experimentally. Therefore, a mathematical model that further improves themodel used in the present study should be constructed.In conclusion, a problem regarding the FDP, namely, a firing rate abstraction, in which the temporal average of spikes istaken, is discussed, ignoring a large amount of extra information within the encoding window, such as the variation offiring pattern [3, 25, 49]. This loss of information contrasts the encoding of rapidly changing neuronal activity observedin the brain [3, 25]. The present study showed theoretically that the output of synaptic plasticity in neurons receivingthe same input frequency differs depending on the input pattern, the calcium time constant, and the background activity,which are related by neuron type and the anatomical and physiological condition in the brain. This finding suggeststhat information neglected in the view that only the firing rate induces the synaptic plasticity is also involved in thesynaptic plasticity and neural coding. In the future, the ratio at which this information is related to synaptic plasticityand neural coding should be verified experimentally and theoretically.10
PREPRINT - J
ANUARY
14, 2020
We used a model for the FDP based on the calcium control hypothesis of Shouval et al., assuming that the change ofthe synaptic weight is fully determined by the postsynaptic calcium level [34, 62]. This model has been confirmedto integrate STDP observed in acute hippocampal slices within a single theoretical framework [63]. Among the fewstudies that have analytically solved this hypothesis, Yeung et al. [64] calculated the mean values of the calcium tran-sients evoked by a spiking neuron. In the present study, we analytically derived the intracellular calcium concentrationand synaptic weight with respect to the input frequency focusing only on the long-term behavior of the intracellularcalcium concentration and synaptic weight.We incorporated in vivo effects into the model as follows. First, in order to investigate the FDP in vivo, we focused onthree types of firing pattern that are widely observed in the brain: constant-ISI (inter-spike intervals) inputs, Poissoninputs, and gamma inputs. Next, calcium decay time constant of in vivo neurons differs from cell to cell. Previousreports suggested that pyramidal neurons in superficial layers possess faster calcium dynamics than deep layers. Inorder to study the association of the calcium decay time constant with the FDP, we examined two kinds of neuronswith time constants of 40 ms and 80 ms. Finally, in vivo neurons are always subjected to background activity. Thefrequency and magnitude of these neurons depend on the location of the synapse in the brain and the surroundingneuronal activity [30, 33]. Hence, we examined the correlation between the amplitude of background synaptic activityand the FDP.The dynamics of the synaptic weight W ( t ) are governed by ddt W ( t ) = η ( Ca ( t ))[Ω( Ca ( t )) − W ( t )] , (5)where Ca ( t ) represents the intracellular calcium concentration, and η and Ω are functions of intracellular calciumconcentration given by the following formulas: η ( Ca ) = (cid:20) p p Ca ) p + p (cid:21) − , (6) Ω( Ca ) = 0 .
25 + sig( Ca − α , β ) − . Ca − α , β ) , (7)where sig( x, β ) = exp( βx ) / [1 + exp( βx )] , (8)and we used the following parameters: p . s, p p / − , p , p s, α = 0 . µ mol / dm , α = 0 . µ mol / dm and β = β = 80 µ mol / dm [34, 62].The dynamics of the intracellular calcium concentration are described as follows: ddt Ca ( t ) = I NMDA ( t ) − τ ca Ca ( t ) , (9)where τ ca is the calcium decay time constant. In order to investigate the relation between the calcium dynamics andthe synaptic plasticity, we examined two kinds of neurons with time constants of 40 ms and 80 ms, which are knownas representative values in pyramidal cells in the deep cortex (layers V to VI) and the superficial cortex (layers II toIV) [35, 36].In Eq. (9), I NMDA represents the calcium current via the NMDA receptor and is expressed as a function of time andpostsynaptic potential as follows: I NMDA ( t, V ) = H ( V ) h I f Θ( t ) e ( − t/τ f ) + I s Θ( t ) e ( − t/τ s ) i . (10)Here, Θ( t ) is the Heaviside step function and we choose the parameters I f = 0 . , I s = 0 . , τ f = 50 ms, and τ s = 200 ms, and H ( V ) is given by H ( V ) = − P G NMDA ( V − V r )1 + ( M g/ .
57) exp( − . V ) , (11)11 PREPRINT - J
ANUARY
14, 2020where we choose the parameters P = 0 . , G NMDA = − / µ mol · dm − / (m · mV) , M g = 3 . , and a reversalpotential for calcium ions of V r = 130 mV [34]. Since H ( V ) increases monotonically with the membrane potential V before reaching a plateau at V = 27 . , the higher the membrane potential the greater the calcium currentthrough the NMDA receptor, I NMDA , as long as
V < . .The postsynaptic membrane potential is given as the sum of the resting membrane potential V rest , which is set to − mV, and the depolarization terms V epsp + V bg : V ( t ) = V rest + V epsp ( t ) + V bg ( t ) . (12)The depolarization terms in Eq. (12) include both EPSPs generated by binding glutamate to the AMPA receptors( = V epsp ) and background contribution ( = V bg ), which describes the depolarization due to the factors other than EPSP.Here, V epsp is expressed as V epsp ( t ) = X i Θ( t − t i ) h e − ( t − t i ) /τ − e − ( t − t i ) /τ i , (13)where t i indicates the i -th presynaptic spike time, and the time constants are τ = 50 ms and τ = 5 ms [34]. Here, V bg is composed of the summation of the dendritic action potentials, the back propagating action potentials (BPAPs),and the voltage noise applied to the postsynapse. The amplitude of the depolarization generated at the postsynapticdendritic spine by the BPAPs varies, decreasing exponentially with the distance from the soma, at which it is about mV relative to the synapse [32, 65]. The duration of the depolarization by BPAPs also differs among cell types[66]. Moreover, the noise level at dendritic spines has been reported to be similar to that measured at the soma [67].We took these previous studies into consideration in order to perform the numerical simulation and presumed that thespike trains by both BPAPs and voltage noise follow a homogeneous Poisson process. Thus, we simply expressed V bg as follows: V bg ( t ) = s X k Θ( t − t k ) h e − ( t − t k ) /τ − e − ( t − t k ) /τ i , (14)where s = 20 mV and { t k } is a Poisson process with a frequency that varies depending on the simulation conditions.(In all simulations except for those of Figs. 7 and 8, we used a Poisson process with a mean frequency of 1 Hz.) In the present study, we performed numerical simulations as well as analytical calculations in order to investigate theFDP. We used Wolfram Mathematica software in all simulations, and determined the dependence of both the calciumconcentration and the synaptic weight on the stimulation frequency as follows. First, we repeatedly solved Eqs. (5)-(14) numerically as a function of time for each frequency. The calcium concentration as a function of time obtained bythis calculation is similar to the results of a previous paper [64]. Next, after a period of . × ms, which is necessaryfor the system to reach a steady state, the average of the calcium level or the synaptic efficacy between . × msto . × ms was calculated. When simulating with Poisson inputs, we performed the above calculations for atleast three input patterns by changing the random seed, and took the average. The quantitative data are expressed asthe mean of at least three independent experiments plus/minus the standard error of the mean (SEM). In order to investigate the dependence of the postsynaptic calcium concentration on the average presynaptic stimulationfrequency of each input pattern, we developed an approximate analytical solution. By integrating Eq. (9), we canformally express the solution for Ca ( t ) as Ca ( t ) = Z t e τca ( s − t ) I NMDA ( s ) ds . (15)Considering that the ion current through NMDAR ( I NMDA ) is reset to zero each time presynaptic input is applied, Eq.(10) is rewritten as follows for the interval between the presynaptic inputs ˆ t k ≤ s ≤ ˆ t k +1 , where ˆ t k is the time for k -thpresynaptic input ( ˆ t = 0 ms ): 12 PREPRINT - J
ANUARY
14, 2020 I NMDA ( s ) = H ( V ) h I f Θ( s − ˆ t k ) e − ( s − ˆ t k ) /τ f + I s Θ( s − ˆ t k ) e − ( s − ˆ t k ) /τ s i . (16)Now, we make the following assumptions.(Assumption 1) The time dependence of H ( V ) can be neglected because it varies slowly in time compared to theother terms in Eq. (16)(Assumption 2) The spike interval fluctuates stochastically. If we define the average spike interval as ∆ t , ˆ t k is writtenas follows: ˆ t k = δ k ∆ t + ˆ t k − , ˆ t = 0 . (17)Then, ˆ t k = k X k ′ =1 δ k ′ ∆ t ( k ≥ . (18)Inserting Eq. (16) into Eq. (15) with Assumption 1, we obtain Ca ( t ) = H ( V ) N X k =0 Z ˆ t k +1 ˆ t k ds h I f Θ( s − ˆ t k ) e − ( s − ˆ t k ) /τ f (19) + I s Θ( s − ˆ t k ) e − ( s − ˆ t k ) τ s i e τca ( s − t ) = H ( V )[ S fN − ( t ) + T fN ( t ) + S sN − ( t ) + T sN ( t )] , (20)where we have separated the contributions from the N -th presynaptic input, T fN ( t ) and T sN ( t ) from the contributionsfrom the first N − presynaptic inputs, S fN − ( t ) and S sN − ( t ) : S jN − ( t ) := N − X k =0 Z ˆ t k +1 ˆ t k ds I j e − τj ( s − ˆ t k ) e τCa ( s − t ) , ( j = f or s ) (21)and T jN ( t ) := Z t ˆ t N ds I j e − τj ( s − ˆ t N ) e τCa ( s − t ) , ( j = f or s ) . (22)Furthermore, we define S jk +1 ,k ( t ) as S jk +1 ,k ( t ) := Z ˆ t k +1 ˆ t k ds I j e − τj ( s − ˆ t k ) e τCa ( s − t ) = I j τ j e − τCa ( t − ˆ t k ) h e τ j (ˆ t k +1 − ˆ t k ) − i , (23)13 PREPRINT - J
ANUARY
14, 2020where τ f and τ s are defined as follows: τ j := 1 τ Ca − τ j , ( j = f or s ). (24)We write t = ˆ t N + ǫ ∆ t , where ǫ ∆ t represents the time interval between the last spike time ( ˆ t N ) and the time tomeasure the calcium concentration ( t ). Substituting the formula into Eq. (23), we obtain S jk +1 ,k (ˆ t N , ǫ, ∆ t ) := S jk +1 ,k (ˆ t N + ǫ ∆ t )= I j τ j e − τCa ( ǫ ∆ t +ˆ t N − ˆ t k ) (cid:16) e τ j δ k +1 ∆ t − (cid:17) . (25)In the case of ≤ k ≤ N − , we have S jk +1 ,k (ˆ t N , ǫ, ∆ t ) = I j τ j e − τCa ǫ ∆ t (cid:16) e − τj δ k +1 ∆ t − e − τCa δ k +1 ∆ t (cid:17) N Y k ′ = k +2 e − τCa δ k ′ ∆ t . (26)In the case of k = N − , we have S jN,N − (ˆ t N , ǫ, ∆ t ) = I j τ j e − τCa ǫ ∆ t (cid:16) e − τj δ N ∆ t − e − τCa δ N ∆ t (cid:17) . (27)Since we are interested in the long-term behavior of the calcium concentration and synaptic weights, but not in thefluctuations caused by each spike, we take the statistical average over one cycle. Let δ k in Assumption 2 obey theprobability density function ρ ( δ ) . Then the statistical averages of e − τCa δ k ∆ t and e − τj δ k ∆ t can be written as r Ca := D e − τCa δ k ∆ t E = Z ∞ ρ ( δ ) e − τCa δ ∆ t dδ,r j := D e − τj δ k ∆ t E = Z ∞ ρ ( δ ) e − τj δ ∆ t dδ. (28)Hence, the statistical average of Eq. (26) is given as D S jk +1 ,k (ˆ t N , ǫ, ∆ t ) E = I j τ j e − τCa ǫ ∆ t ( r j − r Ca ) r N − k − Ca . (29)Summing from k = 0 to k = N − , the statistical average of Eq. (21) is obtained as D S jN − (ˆ t N , ǫ, ∆ t ) E = I j τ j e − τCa ǫ ∆ t ( r j − r Ca ) 1 − r NCa − r Ca . (30)In order to obtain the long-term behavior of D S jN − E , we take the limit N → ∞ . Since r Ca < , and thus r NCa → as N → ∞ , we obtain D S jN − ( ǫ, ∆ t ) E ≃ I j τ j r j − r Ca − r Ca exp (cid:18) − ǫ ∆ tτ Ca (cid:19) . (31)Similarly, using t = ˆ t N + ǫ ∆ t in Eq. (22) and taking the statistical average, we obtain (in the limit N → ∞ )14 PREPRINT - J
ANUARY
14, 2020 T jN ( ǫ, ∆ t ) = I j τ j (cid:16) e − ǫ ∆ t/τ j − e − ǫ ∆ t/τ Ca (cid:17) . (32)Using Eqs. (31) and (32), we obtain the statistical average of the postsynaptic calcium concentration as h Ca (∆ t, ǫ ) i = H ( V ) X j = f,s I j τ j (cid:20) exp (cid:18) − ǫ ∆ tτ j (cid:19) − − r j − r Ca exp (cid:18) − ǫ ∆ tτ Ca (cid:19)(cid:21) . (33)Furthermore, the statistical average of this equation with respect to the observation time is given by h Ca (∆ t ) i = H ( V ) X j = f,s I j τ j (cid:18) r ′ j − − r j − r Ca r ′ Ca (cid:19) , (34)where r ′ Ca := D e − ǫ ∆ t/τ Ca E and r ′ j := D e − ǫ ∆ t/τ j E . (35) First, we calculate r Ca , r j , r ′ Ca , and r ′ j for the constant-ISI input, which are denoted as r Ca ; c , r j ; c , r ′ Ca ; c , and r ′ j ; c ,respectively. In this case, the probability density function is given by ρ Ca ; c ( x ) = ρ j ; c ( x ) = δ (1 − x ) . Using thisfunction in Eq. (28), we obtain r Ca ; c = e − ∆ tτCa , r j ; c = e − ∆ tτj . (36)Since it is assumed that the sampling time follows a uniform distribution, r ′ Ca and r ′ j are expressed as follows: r ′ Ca ; c = τ Ca ∆ t (cid:16) − e − ∆ tτCa (cid:17) , r ′ j ; c = τ j ∆ t (cid:16) − e − ∆ tτj (cid:17) . (37)Using Eqs. (36) and (37), we obtain the statistical average of the postsynaptic calcium concentration as a function ofthe spike interval ∆ t as follows: h Ca (∆ t ) i = H ( V ) τ Ca ∆ t X j = f,s I j τ j (cid:20) − exp (cid:18) − ∆ tτ j (cid:19)(cid:21) . (38)Note that H ( V ) is a slowly changing and monotonically increasing function of the membrane potential in the vicinityof the resting membrane potential ( − ), and the duration of depolarization by EPSP is approximately 50 to 100ms at most. Therefore, the increase in the average membrane potential remains at approximately . , even in thecase of the highest frequency, e.g., 100 Hz. The average membrane potential, moreover, increases linearly with thestimulation frequency. Thus, H ( V (∆ t )) is approximately expressed as a quadric function of / ∆ t (= f ) . With thisapproximation, we obtain the following expression: h Ca c ( f ) i = τ Ca f ( γ + γ f + γ f ) X j = f,s I j τ j (cid:20) − exp (cid:18) − τ j · f (cid:19)(cid:21) . (39)Here, γ = 1 . × − mV , γ = 3 . × − mV · ms , and γ = 3 . × − mV · m . These values are determinedby finding the relation between the input frequency and the time average of V ( t ) in Eq. (12) and by substituting theobtained values into the quadratic approximation of H ( V ) .15 PREPRINT - J
ANUARY
14, 2020
The time interval of the spike sequence according to the Poisson process follows an exponential distribution, theprobability density function of which is given by ρ Ca ; poi ( x ) = ρ j ; poi ( x ) = e − x . Then, we can calculate r Ca and r j for the Poisson input as r Ca ; poi = τ Ca τ Ca + ∆ t , r j ; poi = τ j τ j + ∆ t . (40)Since the spike interval fluctuates stochastically in the Poisson input, the observation time is considered to fluctuatewith the same statistics. Then, r ′ Ca and r ′ j in the Poisson input, written as r ′ Ca ; poi and r ′ j ; poi , are equal to r Ca ; poi and r j ; poi , respectively. Substituting r Ca ; poi , r j ; poi , r ′ Ca ; poi , and r ′ j ; poi , we obtain the statistical average of the postsynapticcalcium concentration receiving Poisson input as a function of the average frequency as follows: h Ca poi ( f ) i = τ Ca ( γ + γ f + γ f ) X j = f,s I j τ j fτ j f + 1 . (41) The time interval of the spike sequence according to the gamma process follows a gamma distribution, the generalformula for the probability density function of which is given as ρ Ca ;Γ ( x ; α ) = ρ j ;Γ ( x ; α ) = 1Γ( α ) x α − e − x , (42)where α is the shape parameter, and Γ is the gamma function, which is given by Γ( α ) := Z ∞ t α − e − t dt. (43)Since, as in the Poisson input, the spike interval and the sampling time fluctuate with the same statistics, r Ca = r ′ Ca =: r Ca :Γ and r j = r ′ j =: r j :Γ in Eq. (33). Thus, we obtain r Ca :Γ = (cid:18) τ Ca τ Ca + ∆ t (cid:19) α , r j :Γ = (cid:18) τ j τ j + ∆ t (cid:19) α . (44)Noting that the average spike interval of the gamma distribution input is α ∆ t , we can express the statistical average ofthe postsynaptic calcium concentration with gamma process input as follows: h Ca Γ ( f ) i = ( γ + γ f + γ f )( αf ) α X j = f,s I j τ j (cid:16) τ j ατ j f +1 (cid:17) α − (cid:16) τ Ca ατ Ca f +1 (cid:17) α − (cid:16) ατ Ca fατ Ca f +1 (cid:17) α . (45) According to the calcium control hypothesis reported by Shouval et al., the time derivative of the synaptic efficacy W is expressed as a function of intracellular calcium concentration as indicated in Eqs. (5)-(7) [34]. Equation (5)indicates that the synaptic strength approaches an asymptotic value Ω( Ca ( t )) with time constant /η ( Ca ( t )) . Thefunctional form of Ω( Ca ( t )) in Eq. (7) is based qualitatively on the notion that a moderate rise in calcium leads to adecrease in the synaptic weight, whereas a large rise leads to an increase in the synaptic weight. This notion is closelyrelated to the BCM theory, which states that weak synaptic input activity results in a decrease in synaptic strength,16 PREPRINT - J
ANUARY
14, 2020whereas strong input leads to an increase in synaptic weight [4, 68].Although it is difficult to find the exact relation between the synaptic weight W and the stimulation rate f analytically,we can obtain an approximate relation by assuming that W ( t ) converges to a stationary solution in the macroscopictime scale, i.e., lim t →∞ W ( t, f ) := h W ( f ) i ≈ h Ω( Ca ( f )) i . (46)In order to calculate h Ω( Ca ( f )) i , we express the postsynaptic calcium concentration as Ca (∆ t, x, ǫ | r Ca , r j ) ≈ lim N →∞ H ( V ) X j = f,s " T j + S jN,N − + N − X k =0 S jk +1 ,k = H ( V ) X j = f,s I j τ j h e − τj ǫ ∆ t − e − τCa ǫ ∆ t + e − τCa ǫ ∆ t (cid:16) e − τj x ∆ t − e − τCa x ∆ t (cid:17) + e − τCa ( x + ǫ )∆ t (cid:18) r j − r Ca − r Ca (cid:19)(cid:21) , (47)where x = δ N , r Ca , and r j are defined in Eq. (28). By substituting Eq. (47) into the expression for Ω( Ca ) in Eq.(7) and calculating the statistical average with respect to x and ǫ , we obtain an approximate analytical solution for thesynaptic weight as a function of the average input frequency.In the case of the constant-ISI input, the time interval of the spike sequence obeys the probability density function ρ ( x ) = δ (1 − x ) . Moreover, the time interval from the last spike to the sampling time obeys a uniform distribution.Thus, we obtain the statistical average of the synaptic weight as a function of input frequency f as follows: h W c ( f ) i = Z ∞ dx Z dǫ δ (1 − x )Ω( Ca (1 /f, x, ǫ | r Ca ; c , r j ; c )) . (48)In the cases of the Poisson input and gamma process input, the spike interval as well as the time interval betweenthe last spike and the observation time obey exponential and gamma distributions, respectively. Thus, the statisticalaverage of the synaptic weight as a function of input frequency f in these inputs are calculated as follows: h W poi ( f ) i = Z ∞ dx Z ∞ dǫ e − ( x + ǫ ) Ω( Ca (1 /f, x, ǫ | r Ca ; poi , r j ; poi )) , (49) h W Γ ( f ) i = 1Γ( α ) Z ∞ dx Z ∞ dǫ ( xǫ ) α − e − ( x + ǫ ) Ω( Ca (1 /αf , x, ǫ | r Ca ;Γ , r j ;Γ )) . (50)17 PREPRINT - J
ANUARY
14, 2020
REFERENCES [1] S. J. Martin, P. D. Grimwood, R. G. Morris,
Annu Rev Neurosci , 649 (2000).[2] P. J. Sjostrom, G. G. Turrigiano, S. B. Nelson, Neuron , 1149 (2001).[3] F. Weissenberger, M. M. Gauy, J. Lengler, F. Meier, A. Steger, Sci Rep , 4609 (2018).[4] E. L. Bienenstock, L. N. Cooper, P. W. Munro, J Neurosci , 32 (1982).[5] T. V. Bliss, T. Lomo, J Physiol , 331 (1973).[6] T. V. Bliss, A. R. Gardner-Medwin,
J Physiol , 357 (1973).[7] T. V. Bliss, G. L. Collingridge,
Nature , 31 (1993).[8] A. Kirkwood, S. M. Dudek, J. T. Gold, C. D. Aizenman, M. F. Bear,
Science , 1518 (1993).[9] S. M. Dudek, M. F. Bear,
Proc Natl Acad Sci U S A , 4363 (1992).[10] R. M. Mulkey, R. C. Malenka, Neuron , 967 (1992).[11] A. Artola, W. Singer, Trends Neurosci , 480 (1993).[12] W. Gerstner, W. M. Kistler, Spiking neuron models : single neurons, populations, plasticity (Cambridge Univer-sity Press, Cambridge, U.K. ; New York, 2002).[13] S. Song, K. D. Miller, L. F. Abbott,
Nat Neurosci , 919 (2000).[14] G. Q. Bi, M. M. Poo, J Neurosci , 10464 (1998).[15] H. Markram, J. Lubke, M. Frotscher, B. Sakmann, Science , 213 (1997).[16] D. Debanne, B. H. Gahwiler, S. M. Thompson,
J Physiol
507 ( Pt 1) , 237 (1998).[17] D. E. Feldman,
Neuron , 45 (2000).[18] L. I. Zhang, H. W. Tao, C. E. Holt, W. A. Harris, M. Poo, Nature , 37 (1998).[19] M. Graupner, P. Wallisch, S. Ostojic,
J Neurosci , 11238 (2016).[20] W. R. Softky, C. Koch, J Neurosci , 334 (1993).[21] J. J. Knierim, D. C. van Essen, J Neurophysiol , 961 (1992).[22] F. Chance, L. F. Abbott, Simulating in vivo background activity in a slice with the dynamic clamp (Springer,2009), pp. 73–87.[23] G. A. Jacobson, et al. , J Physiol , 145 (2005).[24] M. London, A. Roth, L. Beeren, M. Hausser, P. E. Latham,
Nature , 123 (2010).[25] M. Li, J. Z. Tsien,
Front Cell Neurosci , 236 (2017).[26] J. A. Cummings, R. M. Mulkey, R. A. Nicoll, R. C. Malenka, Neuron , 825 (1996).[27] R. J. Cormier, A. C. Greenwood, J. A. Connor, J Neurophysiol , 399 (2001).[28] K. Cho, J. P. Aggleton, M. W. Brown, Z. I. Bashir, J Physiol , 459 (2001).[29] S. N. Yang, Y. G. Tang, R. S. Zucker,
J Neurophysiol , 781 (1999).[30] R. S. Jones, G. L. Woodhall, J Physiol , 107 (2005).[31] M. Rapp, Y. Yarom, I. Segev,
Proc Natl Acad Sci U S A , 11985 (1996).[32] Y. Bereshpolova, Y. Amitai, A. G. Gusev, C. R. Stoelzel, H. A. Swadlow, J Neurosci , 9392 (2007).[33] A. A. Faisal, L. P. Selen, D. M. Wolpert, Nat Rev Neurosci , 292 (2008).[34] H. Z. Shouval, M. F. Bear, L. N. Cooper, Proc Natl Acad Sci U S A , 10831 (2002).[35] B. Ahmed, J. C. Anderson, R. J. Douglas, K. A. Martin, D. Whitteridge, Cereb Cortex , 462 (1998).[36] Y. H. Liu, X. J. Wang, J Comput Neurosci , 25 (2001).[37] H. E. Speed, L. E. Dobrunz, J Neurophysiol , 799 (2008).[38] N. Brunel, J Comput Neurosci , 183 (2000).[39] M. Deger, M. Helias, C. Boucsein, S. Rotter, J Comput Neurosci , 443 (2012).[40] G. Maimon, J. A. Assad, Neuron , 426 (2009).[41] S. N. Baker, R. N. Lemon, J Neurophysiol , 1770 (2000).18 PREPRINT - J
ANUARY
14, 2020[42] S. D. Burton, N. N. Urban,
J Neurosci , 14103 (2015).[43] M. Li, et al. , Biorxiv p. 145813 (2018).[44] J. J. Moore, et al. , Science (2017).[45] J. L. Carrillo-Medina, R. Latorre,
Front Comput Neurosci , 132 (2016).[46] W. Bialek, F. Rieke, R. R. de Ruyter van Steveninck, D. Warland, Science , 1854 (1991).[47] J. H. S. E. T. M. J. E. S. A. S. E. A. J. H. E. S. M. Eric R. Kandel, Edited,
Principles of Neural Science, 5th Edn (Elsevier Science Publishing Co. Inc., New York, 2013).[48] R. P. N. Rao, B. A. Olshausen, M. S. Lewicki,
Probabilistic models of the brain : perception and neural function/ edited by Rajesh P.N. Rao, Bruno A. Olshausen, Michael S. Lewicki , Neural information processing series (MITPress, Cambridge, Mass., 2002).[49] C. Zhao, et al. , J. Emerg. Technol. Comput. Syst. , 1 (2015).[50] S. Panzeri, N. Brunel, N. K. Logothetis, C. Kayser, Trends Neurosci , 111 (2010).[51] R. Latorre, F. B. Rodriguez, P. Varona, Biol Cybern , 169 (2006).[52] C. Kayser, M. A. Montemurro, N. K. Logothetis, S. Panzeri, Neuron , 597 (2009).[53] E. Hu, et al. , Front Comput Neurosci , 58 (2018).[54] A. Minelli, et al. , Cell Calcium , 221 (2007).[55] D. Jeon, et al. , Neuron , 965 (2003).[56] A. Secondo, et al. , J Biol Chem , 1319 (2015).[57] S. Moriguchi, et al. , Neuropharmacology , 291 (2018).[58] S. Asano, et al. , J Neurochem , 2437 (1995).[59] A. Destexhe, M. Rudolph, D. Pare, Nat Rev Neurosci , 739 (2003).[60] W. C. Stacey, D. M. Durand, J Neurophysiol , 1104 (2001).[61] S. Remy, N. Spruston, Proc Natl Acad Sci U S A , 17192 (2007).[62] H. Z. Shouval, G. Kalantzis,
J Neurophysiol , 1069 (2005).[63] D. Bush, Y. Jin, J Comput Neurosci , 495 (2012).[64] L. C. Yeung, G. C. Castellani, H. Z. Shouval, Phys Rev E Stat Nonlin Soft Matter Phys , 011907 (2004).[65] G. J. Stuart, M. Hausser, Nat Neurosci , 63 (2001).[66] Y. Zheng, L. Schwabe, PLoS One , e88592 (2014).[67] A. Yaron-Jakoubovitch, G. A. Jacobson, C. Koch, I. Segev, Y. Yarom, Front Cell Neurosci , 3 (2008).[68] E. M. Izhikevich, N. S. Desai, Neural Comput15