Transition from asynchronous to oscillatory dynamics in balanced spiking networks with instantaneous synapses
aa r X i v : . [ c ond - m a t . d i s - nn ] S e p Transition from asynchronous to oscillatory dynamics in balanced spiking networkswith instantaneous synapses
Matteo di Volo and Alessandro Torcini
2, 3, 4 Unit´e de Neuroscience, Information et Complexit´e (UNIC),CNRS FRE 3693, 1 avenue de la Terrasse, 91198 Gif sur Yvette, France Laboratoire de Physique Th´eorique et Mod´elisation, Universit´e de Cergy-Pontoise,CNRS, UMR 8089, 95302 Cergy-Pontoise cedex, France Max Planck Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38, 01187 Dresden, Germany CNR - Consiglio Nazionale delle Ricerche - Istituto dei Sistemi Complessi,via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy (Dated: September 5, 2018)We report a transition from asynchronous to oscillatory behaviour in balanced inhibitory networksfor class I and II neurons with instantaneous synapses. Collective oscillations emerge for sufficientlyconnected networks. Their origin is understood in terms of a recently developed mean-field model,whose stable solution is a focus. Microscopic irregular firings, due to balance, trigger sustainedoscillations by exciting the relaxation dynamics towards the macroscopic focus. The same mechanisminduces in balanced excitatory-inhibitory networks quasi-periodic collective oscillations.
Introduction.
Cortical neurons fire quite irregularlyand with low firing rates, despite being subject to a con-tinuous bombardment from thousands of pre-synaptic ex-citatory and inhibitory neurons [1]. This apparent para-dox can be solved by introducing the concept of balancednetwork, where excitation and inhibition balance eachother and the neurons are kept near their firing thresh-old [2]. In this regime spikes, representing the elementaryunits of information in the brain, are elicited by stochas-tic fluctuations in the net input current yielding an ir-regular microscopic activity, while neurons can promptlyrespond to input modifications [3].In neural network models balance can emerge sponta-neously in coupled excitatory and inhibitory populationsthanks to the dynamical adjustment of their firing rates[4–9]. The usually observed dynamics is an asynchronousstate characterized by irregular neural firing joined to sta-tionary firing rates [4, 6, 7, 9]. The asynchronous statehas been experimentally observed both in vivo and invitro [10, 11], however this is not the only state observ-able during spontaneous cortical activity. In particular,during spontaneous cortical oscillations excitation andinhibition wax and wane together [12], suggesting thatbalancing is crucial for the occurrence of these oscillationswith inhibition representing the essential component forthe emergence of the synchronous activity [13, 14].The emergence of collective oscillations (COs) in in-hibitory networks has been widely investigated in net-works of spiking leaky integrate-and-fire (LIF) neurons.In particular, it has been demonstrated that COs emergefrom asynchronous states via Hopf bifurcations in pres-ence of an additional time scale, beyond the one associ-ated to the membrane potential evolution, which can bethe transmission delay [5, 15] or a finite synaptic time[16]. As the frequency of the COs is related to such ex-ternal time scale this mechanism is normally related tofast ( >
30 Hz) oscillations. Nevertheless, despite many theoretical studies, it remains unclear which other mech-anisms could be invoked to justify the broad range ofCOs’ frequencies observed experimentally [17].In this Letter we present a novel mechanism for theemergence of COs in balanced spiking inhibitory net-works in absence of any synaptic or delay time scale. Inparticular, we show for class I and II neurons [18] thatCOs arise from an asynchronous state by increasing thenetwork connectivity (in-degree). Furthermore, we showthat the COs can survive only in presence of irregularspiking dynamics due to the dynamical balance. Theorigin of COs can be explained by considering the phe-nomenon at a macroscopic level, in particular we extendan exact mean-field formulation for the spiking dynam-ics of Quadratic Integrate-and-Fire (QIF) neurons [19] tosparse balanced networks. An analytic stability analysisof the mean-field model reveals that the asymptotic so-lution for the macroscopic model is a stable focus anddetermines the frequency of the associated relaxation os-cillations. The agreement of this relaxation frequencywith the COs’ one measured in the spiking network sug-gests that the irregular microscopic firings of the neu-rons are responsible for the emergence of sustained COscorresponding to the relaxation dynamics towards themacroscopic focus. This mechanism elicits COs throughthe excitation of an internal macroscopic time scale, thatcan range from seconds to tens of milliseconds, yieldinga broad range of collective oscillatory frequencies. Wethen analyse balanced excitatory-inhibitory populationsrevealing the existence of COs characterized by two dis-tinct frequencies, whose emergence is due, also in thiscase, to the excitation of a mean-field focus induced byfluctuation-driven microscopic dynamics.
The model.
We consider a balanced network of N pulse-coupled inhibitory neurons, whose membrane po-tential evolves as τ m ˙ v i = F ( v i ) + I − τ m g X j ∈ pre ( i ) ε ij δ ( t − t j ) (1)where I is the external DC current, g is the inhibitorysynaptic coupling, τ m = 20 ms is the membrane timeconstant and fast synapses (idealized as δ -pulses) are con-sidered. The neurons are randomly connected, with in-degrees k i distributed according to a Lorentzian PDFpeaked at K and with a half-width half-maximum(HWHM) ∆ K . The elements of the corresponding ad-jacency matrix ε ij are one (zero) if the neuron j is con-nected (or not) to neuron i . We consider two paradig-matic models of spiking neuron: the quadratic-integrateand fire (QIF) with F ( v ) = v [20], which is a current-based model of class I excitability; and the Morris-Lecar(ML) [21, 22] representing a conductance-based class IIexcitable membrane. The DC current and the couplingare rescaled with the median in-degree as I = √ KI and g = g / √ K , as usually done in order to achieve a self-sustained balanced state for sufficiently large in-degrees[4, 6–9, 23, 24]. Furthermore, in analogy with Erd¨os-Renyi networks we assume ∆ K = ∆ √ K . We haveverified that the reported phenomena are not related tothe peculiar choice of the distribution of the in-degrees,namely Lorentzian, needed to obtain an exact mean-fieldformulation for the network evolution [19], but that theycan be observed also for more standard distributions, likeErd¨os-Renyi and Gaussian ones (for more details see theSM [22] and [25]).In order to characterize the network dynamics we mea-sure the mean membrane potential V ( t ) = P Ni =1 v i ( t ) /N ,the instantaneous firing rate R ( t ), corresponding to thenumber of spikes emitted per unit of time, as well asthe population averaged coefficient of variation CV [26]measuring the fluctuations in the neuron dynamics. Fur-thermore, the level of coherence in the neural activity canbe quantified in terms of the following indicator [27] ρ ≡ σ V P Ni =1 σ i /N ! / , (2)where σ V is the standard deviation of the mean mem-brane potential, σ i = h V i i − h V i i and h·i denotes a timeaverage. A coherent macroscopic activity is associatedto a finite value of ρ (perfect synchrony corresponds to ρ = 1), while an asynchronous dynamics to a vanishinglysmall ρ ≈ O (1 / √ N ). Time averages and fluctuationsare usually estimated on time intervals ≃
120 s, afterdiscarding transients ≃ Results.
In both models we can observe collective fir-ings, or population bursts, occurring at almost constantfrequency ν osc . As shown in Fig. 1, despite the almostregular macroscopic oscillations in the firing rate R ( t )and in the mean membrane potential V ( t ), the micro- n e u r on v i ( t ) -30030-1.501.5 V ( t ) -30-20 time (s) R ( t ) ( H z ) time (s) FIG. 1. The panels show (from top to bottom) the raster plotsand the corresponding time traces for the membrane potential v i ( t ) of a representative neuron, for V ( t ) and R ( t ). Left row(black): QIF and right row (blue): ML. The parameter valuesare N = 10000, K = 1000, ∆ = 0 . g = 1 and I = 0 . scopic dynamics of the neurons v i ( t ) is definitely irregu-lar. The latter behaviour is expected for balanced net-works, where the dynamics of the neurons driven by thefluctuations in the input current, however usually the col-lective dynamics is asynchronous and not characterizedby COs as in the present case [4, 6–9, 23, 24].Asynchronous dynamics is indeed observable also forour models for sufficiently sparse networks (small K ),indeed a clear transition is observable from an asyn-chronous state to collective oscillations for K larger thana critical value K c . As observable from Figs. 2 (a,b),where we report the coherence indicator ρ as a func-tion of K for various system sizes from N = 2 ,
000 to N = 20 , ρ vanishes as N − / for K < K c (as we have verified), while it stays finite abovethe transition thus indicating the presence of collectivemotion. This transition resembles those reported forsparse LIF networks with finite synaptic time scales in[28, 29] or with finite time delay in [5, 15]. However,Poissonian-like dynamics of the single neurons has beenreported only in [5, 15].In the present case, in both the observed dynamicalregimes the microscopic dynamics remains quite irregu-lar for all the considered K and system size N , as testifiedby the fact that CV ≃ . CV ≥ ρ and CV as afunction of the external current I and of the parametercontrolling the structural heterogeneity, namely ∆ . Theresults of these analysis are shown in Figs. 2 (c) and(d) for the QIF and for N = 2 , K -2 -1 ρ K -2 -1 ρ I CV ∆ CV CV a) b)c) d) FIG. 2. Upper panels: order parameter ρ versus K for QIF(a) and ML (b), the inset report the corresponding CVs. Thelower panels display in the upper part ρ and in the lower onethe CV versus I (c) and ∆ (d) for the QIF. The data referto various system sizes: namely N = 2000 (black), 5000 (red),10,000 (green) and 20,000 (violet). The employed parametersare I = 0 . g = 5, and ∆ = 1 for ML (b); for QIF g = 1,∆ = 0 . I = 0 . K = 1000. In both cases we fixed a in-degree
K > K c in order toobserve collective oscillations and then we increased I or ∆ . In both cases we observe that for large I (∆ )the microscopic dynamics is now imbalanced with fewneurons firing regularly with high rates and the major-ity of neurons suppressed by this high activity. This in-duces a vanishing of the CV , which somehow measuresthe degree of irregularity in the microscopic dynamics.At large I the dynamics of the network is controlledby neurons definitely supra-threshold and the dynam-ics becomes mean-driven [30, 31]. The same occurs byincreasing ∆ , when the heterogeneity in the in-degreedistribution becomes sufficiently large only few neurons,the ones with in-degrees in proximity of the mean K , canbalance their activity, while for the remaining neurons itis no more possible to satisfy the balance conditions, asrecently shown in [32–34]. As a result, COs disappear assoon as the microscopic fluctuations, due to the balancedirregular spiking activity, vanish. Effective Mean-Field Model.
In order to understandthe origin of these macroscopic oscillations we consideran exact macroscopic model recently derived in [19]for fully coupled networks of pulse-coupled QIF withsynaptic couplings randomly distributed according to aLorentzian. The mean-field dynamics of this QIF net-work can be expressed in terms of only two collectivevariables (namely, V and R ), as follows [19]:˙ R = Rτ m (cid:18) V + Γ π (cid:19) , ˙ V = V + Iτ m + R ¯ g − ( πR ) τ m (3) where ¯ g is the median and Γ the HWHM of theLorentzian distribution of the synaptic couplings.Such formulation can be applied to the sparse networkstudied in this Letter, indeed the quenched disorder inthe connectivity can be rephrased in terms of a randomsynaptic coupling [35]. Namely, each neuron i is subjectto an average inhibitory synaptic current of amplitude g k i R/ ( √ K ) proportional to its in-degree k i . Thereforewe can consider the neurons as fully coupled, but withrandom values of the coupling distributed as a Lorentzianof median ¯ g = − g √ K and HWHM Γ = g ∆ . Themean-field formulation (3) takes now the expression: τ m ˙ R = R (2 V + g ∆ π ) (4) τ m ˙ V = V + √ K ( I − τ m g R ) − ( πRτ m ) . (5)As we will verify in the following, this formulation rep-resents a quite good approximation of the collective dy-namics of our network. Therefore we can safely employsuch effective mean-field model to interpret the observedphenomena and to obtain theoretical predictions for thespiking network.Let us first consider the fixed point solutions ( ¯ V , ¯ R ) ofEqs. (4,5). The result for the average membrane poten-tial is ¯ V = ( − g ∆ ) / (2 π ), while the firing rate is givenby the following expression¯ Rτ m = g √ K π s π √ K I g + ∆ K − ! . (6)This theoretical result reproduces quite well with the sim-ulation findings for the QIF spiking network in the asyn-chronous regime (observable for sufficiently high ∆ and I ) over a quite broad range of connectivities (namely,10 ≤ K ≤ ), as shown in Fig.3 (a). At the leadingorder in K , the firing rate (6) is given by R a τ m = I /g ,which represents the asymptotic result to which thebalanced inhibitory dynamics converges for sufficientlylarge in-degrees irrespectively of the considered neuronalmodel, as shown in Fig.3 (a) and (b) for the QIF andML models and as previously reported in [24] for LeakyIntegrate-and-Fire (LIF) neurons. In particular, for theML model the asymptotic result R a is attained alreadyfor K ≥ are required.The linear stability analysis of the solution ( ¯ V , ¯ R ) re-veals that this is always a stable focus, characterizedby two complex conjugates eigenvalues with a nega-tive real part Λ R τ m = − ∆ / π and an imaginary partΛ I τ m = q Rτ m (2 π ¯ Rτ m + √ Kg ) − (∆ / π ) . Thefrequency of the relaxation oscillations towards the stablefixed point solution is given by ν th = Λ I / π . This repre-sents a good approximation of the frequency ν osc of thesustained collective oscillations observed in the QIF net-work over a wide range of values ranging from ultra-slowrhythms to high γ band oscillations, as shown in Fig. K < R > ( H z ) K K -4 -3 -2 -1 I ν o sc ( H z ) -2 -1 a) b)d)c) FIG. 3. Upper panels: average firing rates h R i versus K forQIF (a) and ML(b), the horizontal dashed (magenta) lines de-note R a and the solid (red) line in (a) ¯ R in Eq. (6). The choiceof parameters ( I , ∆ ) sets the dynamics as asynchronous:(1 ,
3) in (a) and (0 . ,
8) in (b). Lower panels: ν osc versus I (c) and versus K (d) for the QIF, the insets display the samequantities for the ML. The red solid line in (c) refers to ν th ,and in (d) to the theoretically predicted scaling ν th ∼ K ;the red dashed line in the inset of (c) and (d) to power-lawfitting ν osc ≃ I . and ν osc ≃ K . , respectively. Oscilla-tory dynamics is observable for the selected parameter’s val-ues ( I , ∆ ): (0 . , .
3) in (c) and (0 . , .
5) in (d). Otherparameters’ values N = 10 , g = 1, and K = 1000 in (c). ν th predictsthe correcting scaling of ν osc for the QIF for sufficientlylarge DC currents and/or median in-degree K , namely ν th ≈ I / K / (as shown in Fig. 3 (c) and (d)). Forthe ML we observe similar scaling behaviours for ν osc ,with slightly different exponent, namely ν osc ≈ I . and ν osc ≃ K . , however in this case we have not a theoret-ical prediction to compare with (see the insets of Fig. 3(c) and (d)). Excitatory-inhibitory balanced populations
So far wehave considered only balanced inhibitory networks, butin the cortex the balance occurs among excitatory andinhibitory populations. To verify if also in this case col-lective oscillations could be identified we have considereda neural network composed of 80% excitatory QIF neu-rons and 20% inhibitory ones (for more details on theconsidered model see the SM in [22]). The analysis re-veals that also in this case collective oscillations can beobserved in the balanced network in presence of irregu-lar microscopic dynamics of the neurons. This is evidentfrom the raster plot reported in Fig. 4 (a). An importantnovelty is that now the oscillations are characterized bytwo fundamental frequencies as it becomes evident fromthe analysis of the power spectrum S ( ν ) of the mean volt-age V ( t ) shown in Fig. 4 (b). As expected for a noisyquasi-periodic dynamics, the spectrum reveals peaks offinite width at frequencies that can be obtained as linearcombinations of two fundamental frequencies ν and ν .The origin of the noisy contribution can be ascribed to time (s) n e u r on ν ( H z ) I ν ( H z ) ν (Hz) S ( ν ) a) c)d) ν ν ν - ν ν + ν ν b) ν + ν FIG. 4. The raster plot for a network of N E = 20 ,
000 exci-tatory (green) and N I = 5 ,
000 inhibitory (red) QIF neuronsis displayed in (a). In (c) and (d) the COs’ frequencies, mea-sured from the power spectrum S ( ν ) of the mean voltage V ( t )(shown in (b)), are reported as symbols versus the excitatoryDC current I e . The dashed lines are the theoretical mean-field predictions. The values of the parameters are reportedin [22]. the microscopic irregular firings of the neurons. Anal-ogously to the inhibitory case, a theoretical predictionfor the collective oscillation frequencies can be obtainedby considering an effective mean-field model for the ex-citatory and inhibitory populations of QIF neurons. Themodel is now characterized by 4 variables, i.e. the meanmembrane potential and the firing rate for each popula-tion, and also in this case one can find as stationary solu-tions of the model a stable focus. However, the stabilityof the focus is now controlled by two couples of complexconjugate eigenvalues, thus the relaxation dynamics ofthe mean-field towards the fixed point is quasi-periodic(see the SM for more details [22]). A comparison betweenthe theoretical values of these relaxation frequencies andthe measured oscillation frequencies ν and ν associatedto the spiking network dynamics is reported in Figs. 4(c) and (d) for a wide range of DC currents, revealing anoverall good agreement. Thus suggesting that the mech-anism responsible for the collective oscillations remainsthe same identified for the inhibitory network. Conclusions.
We have shown that in balanced spikingnetworks with instantaneous synapses COs can be trig-gered by microscopic irregular fluctuations, whenever theneurons will share a sufficient number of common inputs.Therefore, for a sufficiently large in-degree the erraticspiking emissions can promote coherent dynamics. Wehave verified that the inclusion of a small synaptic timescale does not alter the overall scenario [25].It is known that heuristic firing-rate models, character-ized by a single scalar variable (e.g. the Wilson-Cowanmodel [36]), are unable to reproduce synchronization phe-nomena observed in spiking networks [37, 38]. In thisLetter, we confirm that the inclusion of the membranedynamics in the mean-field formulation is essential tocorrectly predict the frequencies of the COs, not onlyfor finite synaptic times (as shown in [38]), but also forinstantaneous synapses in dynamically balanced sparsenetworks. In this latter case, the internal time scaleof the mean-field model controls the COs’ frequenciesover a wide and continuous range. As we have veri-fied, sustained oscillations can be triggered in the mean-field model by adding noise to the membrane dynamics.Therefore, an improvement of the mean-field theory herepresented should include fluctuations around the meanvalues. A possible strategy could follow the approach re-ported in [37] to derive high dimensional firing-rate mod-els from the associated Fokker-Planck description of theneural dynamics [5, 15]. Of particular interest would beto understand if a two dimensional rate equation [37]is sufficient to faithfully reproduce collective phenomenaalso in balanced networks.Our results pave the way for a possible extension of thereported mean-field model to spatially extended balancednetworks [39–42] by following the approach employed todevelop neural fields from neural mass models [43].The authors acknowledge N. Brunel for extremely use-ful comments on a preliminary version of this Letter, aswell as V. Hakim, E. Montbri´o, L. Shimansky-Geier forenlightening discussions. AT has received partial supportby the Excellence Initiative I-Site Paris Seine (No ANR-16-IDEX-008) and by the Labex MME-DII (No ANR-11-LBX-0023-01). The work has been mainly realizedat the Max Planck Institute for the Physics of ComplexSystems (Dresden, Germany) as part of the activity ofthe Advanced Study Group 2016/17 “From Microscopicto Collective Dynamics in Neural Circuits”. [1] A. Destexhe and D. Par´e, Journal of neurophysiology ,1531 (1999).[2] T. P. Vogels, K. Rajan, and L. F. Abbott, Annu. Rev.Neurosci. , 357 (2005).[3] S. Den`eve and C. K. Machens, Nature neuroscience ,375 (2016).[4] C. van Vreeswijk and H. Sompolinsky, Science , 1724(1996).[5] N. Brunel, Journal of Computational Neuroscience , 183(2000).[6] A. Renart, J. de la Rocha, P. Bartho, L. Hollender,N. Parga, A. Reyes, and K. D. 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