EEGs disclose significant brain activity correlated with synaptic fickleness
EEEGs disclose significant brain activity correlated with synaptic fickleness
Jorge Pretel, Joaquín J. Torres, ∗ and J. Marro Institute “Carlos I” for Theoretical and Computational Physics, University of Granada, Spain
We here study a network of synaptic relations mingling excitatory and inhibitory neuron nodesthat displays oscillations quite similar to electroencephalogram (EEG) brain waves, and identifyabrupt variations brought about by swift synaptic mediations. We thus conclude that correspond-ing changes in EEG series surely come from the slowdown of the activity in neuron populationsdue to synaptic restrictions. The latter happens to generate an imbalance between excitation andinhibition causing a quick explosive increase of excitatory activity, which turns out to be a (first-order) transition among dynamic mental phases. Besides, near this phase transition, our modelsystem exhibits waves with a strong component in the so-called delta-theta domain that coexistwith fast oscillations. These findings provide a simple explanation for the observed delta-gamma and theta-gamma modulation in actual brains, and open a serious and versatile path to understanddeeply large amounts of apparently erratic, easily accessible brain data.
Today one successfully associates most brain activitywith events in which large sets of neurons cooperate ar-bitrated by willful variations of their synaptic relations[1]. This broadcasts signals throughout, and EEG explo-ration on the cerebral cortex has thus become a relativelysimple, convenient and inexpensive way of analyzing con-sequences of such an intriguing collaboration [2–5]. Infact, EEG studies deliver some overall image of the brainactivity with good time accurateness that complementsmagnetic resonance analysis of better spatial resolution.Specifically, EEGs watch over frequencies and often dis-tinguish δ, θ, α, β and γ “rhythms” —subsequently alongthe range 0.5 Hz to 35 Hz and more—, which are looselyassociated to different states of consciousness such as say,deep sleep, anesthesia, coma, relax, and attention.Truly, this is at present a main noninvasive tool todeepen on the brain operation under both normal andpathological conditions [3, 6–9], and it is therefore con-venient digging out on the interpretation of all of thosewaves details. Indeed, a number of prototypes have al-ready addressed the origin and nature of observed brainoscillatory behavior, e.g. [2, 10–13]. Recently, following ahint [10] that α rhythms might come out from filtering ofcooperative signs by interactions with noisy signals fromdifferent parts of the nervous system, it was explainedthe emergence of a wide spectrum of brain waves within asimple computational framework [14]. More specifically,this study has shown that a neural module can exhibitwaves in a variety of frequency bands just by tuning theintensity of a noisy input signal. We interpret this resultas suggesting that a unifying mechanism in some wayoccurs at some level of brain activity for a range of oscil-lations. In fact, existing literature by now has described[1] various well defined, let us say, dynamic phases , aswell as transitions among them —typically, from stateswith a low and incoherent activity to others that showa high synchrony— where weak signals are processed ef-ficiently in spite of much unrests around. One thus un-derstands, for instance, that this ability is due to thevery large susceptibility developed in the medium by a phase transition due to a mechanism occasionally termedas stochastic resonance [15].The picture in previous theoretically-oriented EEGwork, including [14], is mostly phenomenological and gen-erally adopts a uniform and stationary description of theneuron relations efficiency, thus forgetting the actual pos-sibility that synapses perform dynamically during theneuron cooperation processes [1, 16–18]. Nevertheless,synaptic relations surely vary with time while affectingessentially both the neuron network global behavior andthe ensuing capacities to transmit information [1, 18–23]. For instance, a sort of sudden synaptic facilitationcan allow for transient persistent activity after removalof a stimulus [22], which may be the basis for work-ing memory. Moreover, It was reported that synapticdynamics may induce in the human cortex bursting es-corted of asynchronous activity [18], as well as instabil-ities prompting transitions among attractors, which al-low for effective memory searching [23], in addition toa kind of ‘up-and-down states’ reported to occur in cor-tical neuron populations [24]. Aware of these and sim-ilar facts, we mathematically recast and generalize hereboth the more mesoscopic description in [2] and the al-gorithmic model in [14], perfecting them with detaileddynamic synapses and other realistic features. We thusshow how certain levels of short-time ’depression’ of thesynaptic links induce transitions between states of syn-chronized excitatory-inhibitory neuron populations andglobal states of incoherent behavior. It follows that onemay speak of kind of sharp phase transitions, clearly dis-playing metastability and hysteresis, that have been ex-perimentally observed [25]. Furthermore, near such ex-plosive variations, our model exhibits oscillations with aprominent component in the δ − θ band along with highfrequency activity, namely, the δ − γ and θ − γ modula-tions already perceived in actual brain EEG recordings[26, 27], which have been associated with fluid intelli-gence [28]. Even more, we here associate such intriguingsharp variations with disruptions of the balance amongexcitation and inhibition produced by depression of ex- a r X i v : . [ q - b i o . N C ] F e b citatory inputs into inhibitory neurons. This reduces theinhibitory activity thus prompting a sudden excitationincrease that further reduces inhibition. Interestinglyenough, a lack of the excitation-inhibition balance in theactual human brain could be crucial to understand theessentials of some recurrent neurological disorders suchas epilepsy, autism and schizophrenia, e.g., [29, 30].The simplest version of our model aims to capture theessentials of the cerebral cortex operation allowing fora network with excitatory (E) and inhibitory (I) neu-rons, the former occurring four times the latter. Fur-thermore, the amplitude of the corresponding postsynap-tic responses follows the opposite ratio, i.e., the responseevoked by any I is four times larger than that by any E.This is supposed to correspond to a realistic cortex bal-anced state [31, 32]. We then represent a region of thecerebral cortical tissue with a large square of N nodeswith periodic boundary conditions and fulfilling such abalance, in which each I node influences a set of 12 neigh-boring E’s and it is influenced by 32 adjacent E’s. As inprevious work [10, 14], we do not consider here E-E andI-I feedbacks. Besides, from the various familiar types ofimaginable neuron dynamics, we refer to the celebratedintegrate-and-fire case [1, 33]. Namely, the cell membraneacts as a capacitor subject to several currents, which re-sults in a potential V for each neuron changing with timeaccording to τ dVdt = − V + V in + V noise . (1)Here, as in previous work [10, 14], the time constant τ is set equal to τ ( τ ) according the membrane cell volt-age is above (below) certain resting potential, which weset to zero. The last two terms in (1) correspond to thevoltage induced by the sum of all currents through themembrane, which we separate here in two main contri-butions. V in is the sum of inputs from adjacent neigh-bors that influence the given cell, while V noise accountsfor any input from neurons in other regions of the brain.Assuming lack of correlations [34], we represent V noise asa Poisson signal characterized by a noise level parameter µ . It is now well established that, in human brains,synapses linking neurons may undergo variations in scalesfrom milliseconds to minutes, in addition to more famil-iar long-term plastic effects. In fact, one observes short-term depression (STD), in which the synaptic efficacy de-creases due to depletion of neurotransmitters inside the synaptic button after heavy presynaptic activity [16]. Inaddition, there was reported kind of short-term facilita-tion characterized by an increase of the efficacy strength[35–37], which results from a growth of the intracellularcalcium concentration after the opening of the voltagegated calcium channels due to successive arrival of ac-tion potentials to the synaptic button. It seems that, in general, these two short-term mechanisms may compete[1, 21] but, for simplicity, we just consider here synapsesendowed of STD, and describe this by using the releaseprobability U and the fraction of neurotransmitters attime t ready (to be released) after the arrival of an ac-tion potential x t [20]. The ensuing image is that, eachtime a presynaptic spike occurs, a constant portion U ofthe resources x t is released into the synaptic gap, andthe remaining fraction − x t becomes available again atrate /τ rec . Therefore, dx t dt = 1 − x t τ rec − U x t δ ( t − t sp ) , (2)where the delta function makes that the second right-hand term only occurs for t = t sp , the time at which apresynaptic input spike arrives. Assuming also the am-plitude of the response proportional to the neurotrans-mitters fraction released after the input spike, the STDeffect can be written, for E and I neurons respectively, asfollows V in,dt = V d U x t sp [Θ ( t − t sp ) − Θ ( t − t sp − t max )] (3) V in,ht = V h U x t sp Θ ( t − t sp ) e − ( t − t sp ) τ (4)where Θ( X ) is the Heaviside step function. The form ofthese inputs generated by E and I neurons are chosen sothat the response generated on the postsynaptic neuronmembrane matches data; see, for instance, [14]. Thus,for simplicity, we model the excitatory synaptic inputby a square pulse of width t max and maximal amplitude V d , as described by Eq. (3), and the inhibitory input bya decaying exponential behavior with time constant τ and maximum amplitude V h , as in Eq. (4). In addition,to account for synaptic strength variations that dependon presynaptic history, we multiply these input functionsby a factor U · x t sp , thus ensuring that the amplitude ofthe synaptic input is proportional to the amount of neu-rotransmitters released right after a presynaptic spike,which is an activity dependent factor through dynamicsin Eq. (2). Note that there is no synaptic variabilitypresent when U · x t sp = constant occurring for τ rec → . Furthermore, to prevent the membrane potential in (1)from reaching physiologically unrealistic levels, we im-pose upper and lower limits of V sat = 90 mV and V min = − mV , respectively, around the resting mem-brane potential, V rest = 0 as said. This is achievedby multiplying the different excitatory and inhibitory in-puts by the terms ( V sat − V ) /V sat and ( V min − V ) /V min ,respectively.Equations (1)-(4) fully describe the dynamics of themembrane potential in our basic model below a thresholdfor firing, which is in principle set V th = 6 mV abovethe resting membrane potential for both E and I neu-rons. Additionally, after generation of a spike at t f , weassume an absolute refractory period ( t a ) during whichthe neuron is unable to fire again, and a subsequent rela-tive refractory in which the ability to produce new spikesis constrained. Therefore, we set V th ( t ) = (cid:26) V sat t f < t < t f + t a V sat − e − κ ( t − t f − t a ) t > t f + t a . That is, the threshold is first set to V sat (during one hun-dred time steps, which gives t a = 4 ms ) to impede anyfurther spike generation during t a . Then, it decays expo-nentially to its resting value of mV with a time constant κ − = 0 . ms that mimics the existence of a relative re-fractory period.Using this clear-cut and supposedly realistic model,we numerically analyzed how synaptic STD affects even-tually emergent waves by carefully monitoring the net-work dynamics for adiabatically increasing values of thenoise parameter µ . Figure 1 depicts the resulting averagemembrane potential of the E population versus µ , whichclearly illustrates the mentioned sharp transitions. Thatis, in the absence of STD (top panel in each column),waves do not vary essentially with the external noise am-plitude within µ ∈ (0.5, 100), as already reported in [14].This regime corresponds to the simplest and most famil-iar brain waves. However, when STD is on —namely, thesynaptic efficacies vary with the system activity so thatparameter τ rec is large enough— an ‘explosive’ transitionmay show up as µ increases. This occurs at lower τ rec thelower the maximal excitatory postsynaptic amplitude V d is. It is said ‘explosive’ in the sense that the transitionshows hysteresis, from well-defined synchronized behav-ior to a state of high excitation and low coherence, aswe vary µ adiabatically forward (purple line) and back-ward (green line). Ona may also name this a first-orderphase transition by simple analogy with thermodynam-ics, though with the warning that the present setting isa nonequilibrium one [38].The resulting phase diagram in the ( µ, τ rec ) space isillustrated in figure 2. The solid quasi-vertical line, for µ < . , describes a (continuous or second-order) phasetransition between a near silent phase A, with asyn-chronous sporadic spikes at low rate (corresponding tothe asynchronous down state actually observed in thebrain), and an oscillatory phase B, where brain wavesemerge with increasing frequency as µ increases (see fig-ure 1). As τ rec increases in the system, figure 2 indicatesthat the brain waves disappear at a (first-order) transi-tion (dashed line), where a new phase D of waves withhigh excitation and low coherence emerges. This sharptransition becomes smooth above a say ‘tricritical point’(1.4, 268) (short quasi-vertical solid line on top). Thesmall region C shows metastability as revealed by hys-teresis. Note that when µ is large this region C narrows Figure 1. Evidence for sharp changes in emergent cooperative-neuron EEG-like waves as the noise level µ varies when synap-tic depression is set on. Columns are, from left to right, for V d = 8 , and mV , respectively and V h = − V d . Inall cases, U = 0 . , τ = 16 ms , τ = 26 ms and external ex-citatory noisy inputs modelled inducing each one a constantdepolarization V do = 5 . mV . This is for a module with 196E’s and 49 I’s. as noise level increases. In addition, region B contains(red and blue) areas in which brain waves sharply emergewith high values of the firing rates (>100 Hz) for E andI neurons.Trying to deep on the nature of the sharp transition,we monitored (figure 3) the change with depression ofboth the mean firing rate and the mean amplitude of theoscillations in E and I neuron populations when it occurs(for µ = 3). This shows that, as STD increases, E neu-rons induce the I ones to slowly decaying their firing ratesas approaching the transition point, where they becomesilent. A feedback induced by this decay of the I activ-ity makes the E’s to increase their firing activity untilreaching (at the transition point) its maximum possible,then remaining firing at the maximum possible frequency.This induces important facts on the ensuing oscillations:the amplitude of the inhibitory component of the wavesjumps to zero at the transition point, and the amplitudeof the excitatory component decays to a very low valuebelow V th . Also interesting is how the nature of the emergingwaves changes with STD. For a relatively low noise, e.g. µ = 0.8, the network’s response remains nearly unchanged,while the amplitude of the oscillations decays until nowell-defined oscillatory behavior is observed as STD isincreased (figure 4, case µ = 1). Note that this transi- Figure 2. Diagram ( µ, τ rec ) illustrating different (dynamical)phases in our system. For low noise (region A), there arerandom sporadic excitatory firing events unable to depolarizethe I neurons. Region B shows well-defined rhythms rang-ing from α to γ bands, while a higher depression inducesceasing of the inhibitory activity and a consequent absenceof synchronicity and coherence in region D. Metastability asin figure 1 characterizes the region C. Red and blue coloredareas in B indicate emerging waves with high values of the fir-ing rates (>100 Hz) for E and I neurons, respectively. Dashedlines illustrate first-order phase transitions, while continuouslines denote second-order transitions.Figure 3. Left : Average firing rate for E and I neurons as thelevel of STD is increased until the explosive transition occursfor an external depolarizing noise µ = 3 . Right : Correspond-ing average amplitude of the oscillations. This illustrates thatthe transition occurs because of a cease of firing of I’s due tothe negative feedback of highly depressed I’s over E’s, whichthen star to fire at the transition point at the higher frequency,thus depressing even more the I’s until impeding their firing. tion from a state with synchrony to an incoherent onebecome abrupt as described above for a level of noise µ > (see figure 5).For higher values of µ (e.g., µ = 3 in figure 5),the power spectrum of the response shows significantchanges. First, its peak frequency notably increases forhigher levels of STD, becoming up to twice as great asfor the static case ( τ rec = 0 ), thus inducing waves in the β and γ regimes. This STD-induced transition from Figure 4. Emergence of “ α rhythms” (around 10Hz) in themodel for noise levels µ = 0.6, 0.8 and 1.0, respectively, fromleft to right. Although the power of the main frequency ofthe waves decays as STD increases, this illustrates how thesewaves details are not dramatically affected by synaptic de-pression and the α band regime remains until τ rec ≈ ms ,where the waves disappear (note that this transition becomesexplosive for µ (cid:38) as shown in figure 5).Figure 5. Left: Power spectra of the system response as afunction of the recovery time τ rec for µ = 3 . Right: time se-ries of the emergent oscillations for particular levels of synap-tic depression, namely, τ rec = 0, 145, 230 and 265 ms, respec-tively, from top to bottom. The associated power spectra foreach of these series are highlighted (with the same color) inthe surface plot of the left panel. low to high frequency bands confirms that synaptic plas-ticity could be an important mechanism in modulatingthe nature of the oscillations from cortical neuronal pop-ulations. In addition, we observe that an increase of τ rec can produce secondary, low-frequency peaks coexistingwith the main peak in the power spectrum of the emer-gent waves. This phenomenon is most evident near theexplosive transition point, where a prominent componentin the δ/θ bands emerges, accompanied by a general en-hancement in the amplitude of the oscillations, as canbe seen in the time series presented in figure 5. This ef-fect seems to occur for all relatively high levels of noisy,namely, µ > . Concerning the effect of the E/I balance on emer-gent behavior, it interests how it affects the incidenceof δ ( θ ) − γ modulations around the transition, and howthe appearance of this is affected by the level of synap-tic depression. Figure 6 illustrates some effects of theratio between the E and I synaptic efficacies. We ob-serve that, when V d /V h decreases and the inhibitorysynapses become relatively more influential, the low fre-quency δ/θ component becomes more significant for os- Figure 6. Effect of varying the E/I amplitude ratio as de-pression increases.
Top:
Increasing the amplitude of I’swhile leaving the E’s unchanged enlarges the δ/θ compo-nent of the δ ( θ ) − γ modulation around the phase transition( τ rec ≈ ms ). Bottom:
Increasing the E’s while maintain-ing the I’s moves the transition to higher levels of depressionand makes the emergent oscillations more sensitive to synap-tic depression. cillatory behavior (see Figure 6, top-right and bottom-left panels) while, when this ratio increases, the low fre-quency band components ( δ and θ ) tend to disappear(Figure 6, top-left and bottom-right panels). Addition-ally, an increase of V d /V h , which implies more excita-tion, makes the oscillations frequency more susceptible tochanges on synaptic depression (see Figure 6, top-left andbottom-right panels), while a stronger inhibition tends tomaintain the frequency of the emergent waves nearly un-changed against depression increases (Figure 6, top-rightand bottom-left panels).Summing up, we present in this Letter a very simplemodel that, recasting previous EEG related work, hastwo significant features. One is that it provides a well-defined set-up to undertake a systematic interpretationof apparently erratic brain EEG data. These are easilyaccessible today and, as we have foreseen here, happensto carry important information concerning the brains ac-tivity. Furthermore, this model is convenient to admitcomplements that one might suspect to be relevant inthese scenarios such as, for instance, other synaptic mech-anisms, complex synaptic networks and more realisticnode neurons. In addition, and perhaps even more tran-scendental within this context, the framework presentedhere precisely illustrates how the concept of a (nonequi-librium) phase transition [38] may be essential for an ac-curate description of the brain properties.The authors acknowledge financial support from theSpanish Ministry of Science and Technology, and theAgencia Española de Investigación (AEI) under grantFIS2017-84256-P (FEDER funds) and from the Conse- jería de Conocimiento, Investigación Universidad, Juntade Andalucía and European Regional Development Fund,Ref. A-FQM-175-UGR18. ∗ Corresponding author: [email protected][1] J. Marro and J.J. Torres, “Phase Transi-tions in Grey Matter - Brain Architecture andMind Dynamics”,
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