Phase description of spiking neuron networks with global electric and

PHYSICAL REVIEW E 83, 051909 (2011)

Phase description of spiking neuron networks with global electric and synaptic coupling Dipanjan Roy,* Anandamohan Ghosh,† and Viktor K. Jirsa‡ Theoretical Neuroscience Group, Institut des Sciences du Mouvement, UMR6233 CNRS and Université de la Méditerranée, 163 Avenue de Luminy, F-13288 Marseille, France (Received 24 July 2010; revised manuscript received 4 February 2011; published 11 May 2011) Phase models are among the simplest neuron models reproducing spiking behavior, excitability, and bifurcations toward periodic firing. However, coupling among neurons has been considered only using generic arguments valid close to the bifurcation point, and the differentiation between electric and synaptic coupling remains an open question. In this work we aim to address this question and derive a mathematical formulation for the various forms of coupling. We construct a mathematical model based on a planar simplification of the Morris-Lecar model. Based on geometric arguments we then derive a phase description of a network of the above oscillators with biologically realistic electric coupling and subsequently with chemical coupling under fast synapse approximation. We demonstrate analytically that electric and synaptic coupling are differently expressed on the level of the network’s phase description with qualitatively different dynamics. Our mathematical analysis shows that a breaking of the translational symmetry in the phase flows is responsible for the different bifurcations paths of electric and synaptic coupling. Our numerical investigations confirm these findings and show excellent correspondence between the dynamics of the full network and the network’s phase description. DOI: 10.1103/PhysRevE.83.051909

PACS number(s): 87.19.ll, 87.19.lj, 87.19.lm

I. INTRODUCTION

Cortical neurons display various dynamic behaviors including spike train generation, various bursting behaviors, sustained constant frequency firing and adaptive frequency firing [1–4]. Significant theoretical efforts have been devoted to characterizing the biophysical mechanisms that underlie such repetitive activity [5]. In parallel mathematical efforts have been undertaken to capture the dynamical mechanisms underlying this repertoire of neuronal behaviors [6–11]. One of the simplest mathematical representations is the canonical phase model referred to as the theta neuron [12], which can be derived from a saddle node on an invariant circle (SNIC) bifurcation and displays threshold properties and excitable behavior. Beyond the bifurcation point, the membrane voltage displays repetitive firing [14]. SNIC bifurcations lead to Class I excitability observed in various neuron models [10,13,15]. When coupling neurons in a network, previous mathematical studies have used generic local bifurcation arguments to derive the corresponding representation of coupling in the phase description [13,16–21]. No distinction of electric and synaptic coupling can be established through this approach due to the generic nature of the argument. To address this issue of the correct differentiation of electric and synaptic coupling in the phase description, we here take the following approach: We formulate a planar neuron model that allows for a convenient phase description via a coordinate transformation. Under a circular approximation of the phase space portrait, the electric and synaptic couplings assume very distinct and simple characteristic forms, whereas the mathematical expressions characterizing the intrinsic neuronal dynamics are more complicated (that is, they are not as simple as in the theta neuron, for instance). Through a perturbation

*

[email protected] [email protected] ‡ [email protected] †

1539-3755/2011/83(5)/051909(10)

analysis we demonstrate that deviations of the phase portrait from the unit circle do not alter the basic mathematical form of the electric and synaptic phase couplings in the lowestorder approximation. We then numerically confirm that the bifurcation structure of the full spiking neuron network model (using planar as well as Hodgkin-Huxley neuron models) is correctly captured by the reduced phase model. II. NEURON MODEL

As a first step we formulate our neuron model, which shall be based on the Morris-Lecar type biophysical model and related “planar” simplifications [24]. We introduce a cubic nullcline and piecewise (PWL) nullcline. The motivation for this kind of model comes from the reasoning that it can in principle generate type I behavior that is often associated with either a homoclinic or a SNIC bifurcation observed in detailed biophysical models [20,25]. The introduction of the PWL nullcline simplifies the nonlinearity for the gating variables and was previously introduced by Tonnelier et al. [25,26]. Chik et al. have successfully studied clustering through postinhibitory rebound in synaptically coupled neurons using this type of PWL caricature [27]. The governing equations for the single-neuron model are dx = −y − x 3 + x + I dt

(1)

dy = [−y + f (x)]/τ , dt

(2)

and

where

f (x) =

0, for x < x0 , c(x + x0 ) , for x x0 .

(3)

Here, x represents the membrane potential that meets the requirement of a “fast” depolarizing Ca ++ current, and y represents the gating variables that capture the dynamics of 051909-1

©2011 American Physical Society

DIPANJAN ROY, ANANDAMOHAN GHOSH, AND VIKTOR K. JIRSA

a slow repolarizing K + current. The above system, Eqs. (1) and (2), is a two-dimensional planar representation of the dynamics of the Morris-Lecar model [1] in which the fast calcium channels instantaneously assume their steady-state values following a change in the membrane potential. The input current I acts as a bifurcation parameter in our model and moves the nullcline in the vertical direction with no influence on its shape [24]. The parameter values for c = 4I , x0 = −0.25 are chosen such that the nullclines always intersect at the center of the homoclinic orbit as I > Ic . From initial simulations we find that the resulting homoclinic orbit is sufficiently circular. The intrinsic timescale of the slow dynamics is τ ; all numerical studies are carried out for τ > 1. Specifically, here we seek to derive a model, which does not necessarily yield a simple mathematically convenient form (such as the theta neuron model) but rather allows for an explicit formulation of coupling terms. To

f (θ ) =

c

r sin θ+r(θ) cos θ τ

4

aj sin (j θ ) +

j =1

4

bj cos (j θ ) + f (θ ) + O()

j =1

= F (θ ) + O(), f (θ ) =

c(1+cos 2θ ) 2τ

(6) 0 +

cx0 (cos θ) τ

for cos θ < x0 for cos θ x0 ,

− r(θ ) sin(θ ) − r 3 (θ ) cos3 (θ ) + r(θ ) cos(θ )] r sin(θ ) + r(θ ) cos(θ ) [−y0 − r(θ ) sin(θ ) + f (θ )] , + τ (4)

0, [r(θ ) cos θ + x0 ] ,

dr where r = dθ . For sufficiently small enough the dynamics evolves on an invariant circular orbit of unit radius, r(θ ) = 1. Then the above phase equation simplifies to

θ˙ = 0.5 +

this end we make the following ansatz: [x + i(y − y0 )] = r(θ ) exp(iθ ), where x = r(θ ) cos θ and y = y0 + r(θ ) sin θ , where y0 = I . The amplitude correction is defined as r(θ ) = 1 + h(θ ), where is small for near-circular orbits and h(θ ) is the phase dependence of the homoclinic orbit. Substituting this in our governing equations (1)–(3) (we present the detailed calculus in Appendix A) we arrive at the following phase evolution equation for the single neuron: 1 [r cos(θ ) − r(θ ) sin(θ )][(I − y0 ) θ˙ = 2 r + r2

(7)

where the nonlinear flow of the phase is captured by F (θ ), and O() contains all terms of first and higher order in . The parameters are a1 = −(I − y0 ), a2 = −(0.25 + 1/2τ ), a3 = 0, a4 = 0.425, b1 = −y0 /τ , b2 = −0.5, b3 = 0, and b4 = 0. Figure 1 shows the phase plane trajectories obtained for the full model in Eqs. (1)–(3) and the phase model in Eqs. (4)–(5), as well as the circular approximation of the phase model in Eqs. (6)–(7). We express h(θ ) = a0 + a cos (θ ) + b sin (θ ) through a truncated Fourier expansion and determine the coefficients numerically as a0 = −1.5, a = −0.34, b = −0.64. We choose = 0.1, which results in a deviation from circularity of orbit. Simulations of Eqs. (1)–(3) and Eqs. (6)–(7) are shown in Fig. 2. When I < Ic the above system has a pair of fixed points; one is stable and the other is unstable [see Fig. 2(a)]. These are also captured by the phase model as indicated by the vector field in Fig. 2(c). When I ≈ Ic , the two fixed points approach each other, collide, and disappear. For I slightly greater than Ic the trajectory spends practically all its time moving through the bottleneck. Beyond this, for I > Ic , as shown in Figs. 2(b) and 2(d), we are left with an inflection point right at the center of the orbit. As can be appreciated from Figs. 2(c) and 2(d), the nonlinear flow F (θ ) is actually quite simple despite its complicated formulation in Eq. (6). Relevant for a good qualitative correspondence between the dynamics

for r(θ ) cos θ < x0 , (5)

for r(θ ) cos θ x0 ,

of two levels of description is a constant amplitude. This is justified by the phase-plane structure of our model, which can be divided into two main regions of interest. The first region lies in quadrants 2 and 3, and the second in quadrants 1 and 4. In quadrants 1 and 4 where x x0 , using two-time scale analysis, we show that the magnitude of amplitude obeys the = 0.5r[(1 − 1/τ ) − 3r 2 ] [28]. Setting following equation: dr dt the time √ dependence of r to zero and τ 1 results in r ≈ 1/ 3. We verify numerically that this is true for τ 1, where our model shows a stable invariant cycle that is nearly circular. In order to show the existence of a SNIC bifurcation,

1.5 1.0 0.5

y


0 − 0.5 − 1.0 − 1.5 − 1.5 − 1.0 − 0.5

0

0.5

1.0

1.5

x FIG. 1. Phase space dynamics for the planar neuron model and the approximate phase models with and without amplitude correction. The planar model is shown in black dotted curves, the corrected phase model in thick gray dotted curves, and the circular phase model in light gray dotted curves. The two intersecting curves with black square and black circular points are the two nullclines.

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PHASE DESCRIPTION OF SPIKING NEURON NETWORKS . . .


of the couplings in the circular limit will be meaningful if the errors introduced by the noncircular contributions are at least of the order . In this case the zero-order term of coupling will be the leading term. A. Electric coupling

Following is the set of equations that describes a globally electrically coupled system of nonlinear oscillators: dxi = −yi − xi3 + xi + I + K(X − xi ) , dt dyi = [−yi + f (xi )]/τ , dt 0 for xi < x0 f (xi ) = c(xi + x0 ) for xi x0 ,

FIG. 2. Phase space portrait, x, y time series, and nullclines for planar model Eq. (1)–(3) in the excitable [left top panel in (a)] and in the oscillatory [left bottom panel in (b)] regimes. Both dynamical states are linked through a SNIC at Ic = 0.38 (see text for more details). (c) (right top panel) Stability of the fixed points (in black a stable fixed point and in white an unstable fixed point) are depicted in a one-dimensional vector field flow on the real line for the phase model in the excitable regime and in (d) (right bottom panel) in the oscillatory regime. The arrow heads on the real line indicate the phase flow.

we perform a linear stability analysis √ for the full system √ Eqs. (1)–(3) and find that xc = ±1/ 3 and Ic = ∓2/3 3. It is now straightforward to show that near the fixed point, the lowest-order contribution of the local dynamics reduces to √ ( 3I dx 2 ˜ ˜ = I − bx , where I = √3+1) and b = √33 . dt III. NETWORK MODELS

Motivated by the correspondence between the singleneuron model and its approximate circular phase description we now analyze a globally coupled network. Our objective is to evaluate to what degree the here adapted phase description of a network reproduces the qualitative dynamic features and bifurcations as observed in the full network. Kuramoto [22] was among the first to describe how to compute the interaction between the limit cycle oscillators in the limit of infinitesimally weak coupling. By using the method of averaging, it is possible to compute the interaction between the oscillators in the limit when the interactions are vanishingly small. However, this description works well for the limit cycle regime, but not when the neurons are in a nonoscillatory regime. Similar constraints apply to approaches using phase response curves outside of the regular firing regime [23]. Here we take the following approach: We introduce a biologically realistic electrical and synaptic coupling in the neuron model, which is used in many studies to understand clustering, synchronization, etc., in a network of globally coupled oscillators [16,26,29–31]. Then we perform the same transformation as in the previous section for the uncoupled neuron with the goal to derive an expression for the coupling in the limit of circular orbit. The expression

(8) (9) (10)

where i = 1 . . . N and where K is the coupling constant. For an uncoupled network, K = 0. Each neuron receives input from all other neurons, and the average activity of the network is X = 1/N N i=1 xi . The generalized phase equation in the limit of a circular orbit with electric coupling is as follows: θ˙i = F (θi ) − K sin θi (cos − cos θi ) + O(),

(11)

where all parameters in F (θ ) are the same as in (6), and cos = 1/N N l=1 cos θl , i = l. The noncircular contributions are absorbed in O() (see Appendix A). Every neuron is connected to every other neuron in the network, and the coupling strength between any two neurons is K . There is no N self-coupling between the neurons, and the electric coupling constant usually assumes only positive values. However, in this case we extended the parameter space search also to negative (and hence nonbiological) values of K to allow a direct comparison with the effects of synaptic coupling. Results of the simulations of Eqs. (8)–(11) are shown in Fig. 3, which displays four different dynamics in the parameter space as we vary the control parameters (K, I). We compare the four different dynamics obtained from the full system with the phase description of the system for N = 100 oscillators and find good correspondence in the qualitative dynamics (see Fig. 4). We employ the following scheme for the identification of the parameter spaces: First, we determine the saddle-node bifurcation curve by imposing the conditions for phase-coupled networks dθi /dt = 0, det J = 0, where J is the Jacobian matrix; then we solve these simultaneously and sweep θi over the full range from [−π,π ]. Similarly we determine the Hopf bifurcation curve by imposing the conditions dθi /dt = 0, T rJ = 0. For the planar and the phase model composed of identical elements we hold the coupling strength K to be fixed and allow for adjusting I continuously which results in a degenerate point in the (I, K) plane. We fix K to be zero and in the absence of the coupling determine the critical I value for both the full and phase models. Subsequently we adjust K continuously by holding I fixed at its critical value to compute the bifurcation lines limiting synchrony and incoherence. In the numerical bifurcation analysis the degenerate codimension 2 point in the (I, K) plane is detected as a cusp bifurcation point in the parameter plane using forward and backward bifurcation techniques. From simulations we find for both these models starting with random initial phases

051909-3



for more negative values of K (linear coupling strength) population of neurons exhibit bistability and the existence of a separatrix. The increase in coupling strength toward more positive values results in a stable fixed point attractor for both these systems. As we go above I > Ic for fixed coupling strength the systems become oscillatory through a saddle-node bifurcation. Now, depending on the sign of K the entire population of neurons either synchronizes or shows incoherent behavior about their center of mass. This is quite well captured in the simulation as shown in Fig. 3. We numerically verified the robustness of the parameter space structure up to N = 1000 oscillators. The structure of the parameter space is similar to other oscillator types (typically globally coupled phase models close to SNIC bifurcation) like the one proposed by Ref. [16]. To gain insight into the mechanisms underlying the emergence of the oscillatory behavior of the network, we pursue an analytical approach making use of the network’s phase evolution equations θ˙i = F (θi ) − K sin θi (cos − cos θi ),

FIG. 3. Schematic of different scenarios for a population of coupled neurons in the phase space spanned by the variables x and y. Each dot represents the state of a neuron. (a) The population remains in the vicinity of a fixed point attractor. (b) The population shows clustering in the phase space and bistability. (c) The population oscillates out of phase. (d) The population of an oscillator in synchronization. The time series on the right are from a population of 10 neurons for various (I–K) values for the reduced model.

I

where all the parameters are the same as described in Eq. (11). In the parameter space in Fig. 4 the region for K > 0 corresponds to mostly monostable and synchronized solutions, that is, phase-locked solutions where cos θi = cos . For phase-locked solutions the coupling term is essentially zero, ˙ = F (), and hence the and must be a solution of network’s monostable solution changes stability for increasing I at the same time as when the stability change occurs for the single-neuron solution. On the other hand, regions with K < 0 show bistable and incoherent solutions and are characterized by cos = 0; for the bistable case, θi ∈ A, which is the first fixed point θi = 0, and θi ∈ B, which is the second fixed point θi = π ; for the incoherent oscillations, the θi are distributed across the closed orbit. This leads to the following phase evolution equation: θ˙i = F (θi ) + K sin (θi ) cos (θi ).

(a)

Monostable

Monostable

Monostable

-1

0

I

-1

0

Bistable

Bistable

1

Synchrony

0

Incoherence

0.2 0.4 0.6 0.8

(c)

1

Synchrony

0

0 Bistable

K

(b) 1

Synchrony

(13)

I

I

1

(12)

Incoherence

0.2 0.4 0.6 0.8

I

1

-1

0

Incoherence

0.2 0.4 0.6 0.8

1

I

FIG. 4. Parameter space with global electric coupling for (a) phase model, (b) phase model with amplitude correction, and (c) full model. 051909-4


FIG. 5. Time averaged order parameter r(t) t as a function of electric coupling strength K for varying I . Number of spiking neurons N = 100.

The bistable solution may undergo instability through parameter changes of either K or I . When varying K, we consider small perturbations μ to the two fixed-point solutions θi = 0,π . With θi = 0 + μ Eq. (13) becomes θ˙i = μ˙ = F (μ) + K2 sin (2μ), and linearization yields θ˙i = (F (0) + K)μ. Similarly, for θi = π + μ, we can write θ˙i = μ˙ = F (π + μ) + K2 sin (2π + 2μ), and linearization yields θ˙i = [F (π ) + K]μ. With F () ≈ − τI cos from the previous section, we find that F () ≈ τI sin will be generally small for = 0,π . The bistable solution hence becomes unstable when F (0,π ) + K = 0, which defines the almost horizontal critical line between monostability and bistability in Fig. 4. The bifurcation route from bistability to incoherent solutions as the parameter I increases is less conclusive in the framework of the circular approximation, since in the previous stability analysis the only I -dependent term is F (0,π ), which is very small; hence higher orders of the approximation must be considered. From Appendix A we know by definition r(θi ) = 1 + h(θi ). Hence we can compute the nonlinear flow contribution with the first-order correction as F (θi ) + H (θi ). From Eq. (A5) we can determine H (θi ) explicitly and find that H (θi ) has a periodicity of π , that is, H (θ + π ) = H (θ ). Thus the linear stability analysis about the fixed point θi∗ = 0 + μ gives μ˙ = [F (0) + H (0) + K]μ = [F (π ) + H (π ) + K]μ. (14) From the above equation with F (0) ≈ 0 and the π periodicity of H (θi ), we find that the two fixed points at 0,π lose stability at the same time for increasing I , hence there is no route that leads through a region of monostability but leads directly to the incoherent solutions. Since H (θi ) ∼ I scales linearly for fixed μ, we can also estimate the critical line of the parameter space [Figs. 4(a)–4(b)], which separates bistability from incoherence. For the critical line: H (θi ) = m(I − Ic ), where m is the slope of this line and m > 0 allows for destabilization. Hence the critical condition is H (0) + K = 0. By substituting the dependence of H (θi ) on (I,Ic ), one


can write m(I − Ic ) + k = 0. This implies K = −I + Ic for (m > 0). Thus the critical condition for K < 0 is |K| = m(I − Ic ), which serves as a convenient guide to numerically compute the stability line in Figs. 4(a)– 4(b) separating the bistable region from incoherence. In order to prove that the 2 incoherent solution is stable it is sufficient to show drdt < 0 for ∀K < 0, where the Kuramoto order parameter is 1 exp[iθi (t)], (15) r(t) = N i and hence r 2 = [ N1 i cos (θi )]2 + [ N1 i sin (θi )]2 . Our starting equation is Eq. (12). Taking the time derivative of r 2 (t) we obtain the following:

2 2 ˙ 2 ¯ cos (θi ) K sin (θj )(cos θ − cos θj ) r = N i j

2 ¯ + sin (θi ) − K sin(θj ) cos(θj )(cos θ − cos θj ) . N i j (16) The above equation can be further simplified to 2K cos θi cos θ¯ sin2 θj − sin2 θj cos θj r˙2 = N i j j

1 sin 2θj − − sin θi cos θ¯ sin θj cos2 θj . 2 i j j

(17)

¯ = 1 i cos (θi ) we can rewrite By using the fact that cos (θ) N Eq. (17) as 2K 2 2 ˙ sin θj − sin θj cos θj r2 = N cos θ¯ cos θ¯ N j j

1 2 sin 2θj − sin θj cos θj . − N sin θ¯ cos θ¯ 2 j j (18) Moreover, since cos = 0 holds for the incoherent solution, we write cos2 (θj ) sin (θj ). (19) r˙2 = 2K sin j

When we plot the above quantity against the range of negative values of K taking into account sin = 1 and θj ∈ [−π,π ], we find that the expression in Eq. (19) is always negative and hence establishes the stability of the incoherent state. Fully synchronized behavior can be studied for a population of globally coupled oscillators by computing an order parameter defined as in Eq. (15). For a synchronized state r(t) = 1, and for a completely incoherent state r(t) 0 for large N . It is to be noted that there exists intermittent fluctuations near the transition making the order parameter in Eq. (15) greater than zero even for completely incoherent states. The variation of the coupling strength in forward and backward direction leads to no hysteresis. We find a continuous

051909-5


phase transition of the order parameter toward r(t) = 1, a distinct signature for supercritical Hopf transition near which √ the order parameter r(t) ≈ (K − Kc) as shown in Fig. 5.


equation in the limit of a circular orbit with synaptic coupling is as follows: θ˙i = F (θi ) − sin θi (cos θi − vth ) + O(),

(24)

where the mean field for synaptic action for a given population of neurons is expressed as

B. Synaptic coupling

Here we use a globally coupled network model with chemical synapses such as the one proposed by Golomb and Rinzel [32,33]. A single neuron in our model system of N globally synaptically coupled (all-to-all) neurons is similarly described by a set of three nonlinear differential equations: x˙i = −yi − xi3 + xi + I + gS(xi − vth ) , y˙i = [−yi + f (xi )]/τ ,

(20) (21)

s˙i = asi (xi )(1 − si ) − si /β, 0 for xi < x0 f (xi ) = c(xi + x0 ) for xi x0 .

(22)

N g β =

N l=1 1 + β + exp −

,

i = l,

(25)

and higher orders of are absorbed in O(). Figure 6 shows the parameter space diagram for the full and phase models presented in Eqs. (20)–(23) and Eq. (24). Over a wide range of I-g values the collective dynamics of the two systems primarily show four distinct regions of interest, which are close to each other in the parameter space. However, there are qualitative differences in the collective dynamics compared to electric coupling in the previous section. From simulation we find both the full model and the reduced model exhibit bistability and the existence of a separatrix for negative as well as positive values of g (inhibitory and excitatory coupling, respectively). For decreasing coupling strength g, the network displays monostable behavior. Increasing I for fixed values of g results in a phase transition from bistable behavior to monostable behavior. With further increase of the current I > Ic both networks become oscillatory through a saddle-node bifurcation. Now, depending on the coupling strength, the entire population of neurons either synchronizes or shows incoherent behavior about their center of mass. We have obtained numerically the boundaries that separate these four regions in the parameter space. For the entire simulation, we fixed the reversal potential of potassium ions to vth ≈ −1.0. The parameter space structure for both planar and phase models are obtained numerically with standard bifurcation analysis software [38,39]. We test our approximation for fast synapses numerically and find excellent agreement between the planar model and reduced phase model for different values of the time constant β. Following the same lines of thought as we did for the electric coupling, we consider the instability of the bistable

(23)

The collective synaptic action is given by S = N1 N i=1 si , 1 where asi (xi ) = [1+exp(−x . The synaptic constant g is the i /2)] same for all the neurons. In the case of a fast synapse (AMPA-type glutamate receptors), such as those found in the auditory system, the rise time is instantaneous, and postsynaptic responses commence almost instantaneously after the start of presynaptic action potential [34,35]. This brisk communication is a consequence of rapid calcium-channel kinetics, which allows significant calcium entry during the upstroke of the presynaptic action potential [36]. The time course of the postsynaptic conductivity caused by an activation of AMPA receptors can be captured by a rise time βrise = 0.09 ms and decay time βdecay = 1.5 ms [37]. For our simulation we use these values as guidelines. Under the fast synapse approximation the variable si relaxes much more rapidly than xi , in which case we may apply a quasistatic approximation to (22), s˙i ≈ 0, allowing us to adiabatically eliminate the β synaptic variable via si = [1+β+exp(−x . From numerics i /2)] we find this approximation provides good results for τs in the range between 0.01 and 0.5 ms. Then the generalized phase I

cos θl 2

I

I

(a)

(b)

(c)

1 1

0 Incoherence

0 Incoherence

-1

-1 0

0.2

0.4

I

0.6 0.8

Synchrony

Monostable

Monostable

g 0

Synchrony

Bistable

Synchrony

Bistable

Monostable

Bistable

1

1

0

0.2

0.4

I

0.6 0.8

Incoherence

-1 1 0

0.2

0.4 0.6 0.8

1

I

FIG. 6. Parameter space with global synaptic coupling for the (a) phase model, (b) phase model with amplitude correction and (c) full model. 051909-6



dynamic region for increasing parameters I . In particular, we wish to understand why for synaptic coupling the bifurcation path leads us always through the monostable region toward incoherence. The governing equation for the phase network evolution with synaptic coupling reads

reflects gap junction coupling, but does not reflect synaptic coupling. The latter assumes a form of multiplicative type. The two different types of coupling are exact for circular orbits and are valid approximations of the network dynamics for small deviations from circular symmetry. This finding is meaningful, since the resulting qualitative network behaviors are different. The advantage of the phase description of the network dynamics is its simpler mathematical form. Key to the here applied approximation is the assumption of amplitude independence of the couplings. This assumption is trivially satisfied for circular orbits. For more realistic scenarios, there will be a smooth dependence of the amplitude on the phase angle, which is where our approximations hold well. Nevertheless they will fail when the dependencies become too complex, such as in weakly coupled chaotic oscillators. It is important to note that the couplings in the phase descriptions maintain their mathematical expression plus some linearly added correction terms, which scale with the degree of order of deviation from the circle. As a consequence of this additivity of couplings and deviations from circularity, the here presented networks of phase oscillators offer a reasonable simple framework for the study of spiking neurons with mixed couplings and close to circular orbits.

(26) θ˙i = [F (θi ) − sin θi (cos θi − Vth )], g N β where = N j =1 . As observed numerically cos θj [1+β+exp(−

2

)]

the bistable solution has two fixed points θ ∗ = 0,π , where β and subsequently for θ ∗ = 0 implies 0 = g [1+β+exp(− 1 )] 2

β θ ∗ = π , π = g [1+β+exp( 1 . Note that = () and = )] 2

1 1 0 + π = gβ( [1+β+exp(− + [1+β+exp( 1 1 ). In the following 2 )] 2 )] we perform the linear stability analysis around these two fixed points. (a) For θi = 0 + μ we write

sin 2θi (27) θ˙i = μ˙ = F (μ) − + Vth sin θi . 2 Linearization of the above leads to μ˙ = [F (0) − cos(0) + Vth cos(0)]μ, and thus μ˙ = [F (0) − (1 − Vth )]μ = λμ. To allow for a direct comparison with the electric coupling, we consider also noncircular contributions, r(θ ) = 1 + h(θ ) and F (θ ) + H (θ ). This additional contribution extends the stability analysis to θ˙i = μ˙ = [F (0) + H (0) − (1 − Vth )]μ,

(28)

where F (0) ≈ 0, H (0) = H (π ). Then the linear stability is given by λ0 = H (0) − (1 − Vth ). (b) For the second fixed point θi = π + μ in Eq. (26) we write after linearization θ˙i = μ˙ = [F (π ) − cos(2π ) + Vth cos(π )]μ,

(29)

ACKNOWLEDGMENTS

This research has been supported by the ATIP Plus Program (CNRS) and the James S. McDonnell Foundation. APPENDIX A: DEVIATION FROM CIRCULAR APPROXIMATION OF THE HOMOCLINIC ORBIT

The planar network model under global synaptic coupling is governed by the following equations:

which reduces to μ˙ = (F (π ) − (1 + Vth )μ. Again, when extending toward noncircular contributions we obtain

x˙i = −yi − xi3 + xi + I + gS(t)(xi − vth ) ,

(A1)

θ˙i = μ˙ = [F (π ) + H (π ) − (1 + Vth )]μ,

y˙i = [−yi + f (xi )]/τ ,

(A2)

s˙i = asi (xi )(1 − si ) − si /β, 0 for xi < x0 f (xi ) = c(xi + x0 ) for xi x0 .

(A3)

(30)

where F (π ) ≈ 0 and λπ = H (π ) − (1 + Vth ). For Vth > 0 the following inequality is satisfied: 0 > λ0 > λπ which demonstrates that θi = 0 is less stable and destabilizes first for increasing I [note H (θ ) scales linearly in I ]. In other words, the translational symmetry of the phase flow under translation of π is broken for synaptic coupling. As a consequence, the bifurcation path from bistability to incoherence always leads through monostability. IV. SUMMARY

Here we have developed a phase description of globally coupled neurons with fast chemical synapses and biologically realistic electric synapses, which is not constrained to the limit cycle regime but extends to the excitable regime displaying features of multistability and threshold behavior. We demonstrated analytically that the electric and synaptic couplings assume different mathematical forms in the phase description and tested their validity computationally. Notably, the synaptic coupling is expressed in a different mathematical form in the reduced equations than so far assumed in the literature [25,27, 40–43]. The previously assumed coupling expression correctly

(A4)

Using the approximation of adiabatic elimination (see main text) Eq. (A3) assumes the following form: si = β . Now we make the following ansatz: [xi + [1+β+exp(−xi /2)] i(yi − y0 )] = r(θi ) exp(iθi ), where x = r(θi ) cos θi and y = y0 + r(θi ) sin θi , where i = 1 . . . N. Substituting this in our governing equations [Eqs. (A1)–(A3)] with the definition r(θi ) = 1 + h(θi ) allows us to write the phase evolution equation for our network as follows: θ˙i =

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1 [r cos(θi )−r(θi ) sin(θi )][(I − y0 )−r(θi ) sin(θi ) r 2 +r 2 − r 3 (θi ) cos3 (θi ) + r(θi ) cos(θi ) + Isynapse ] +

r sin(θi ) + r(θi ) cos(θi ) [−y0 −r(θi ) sin(θi ) + f (θi )] , τ (A5)


f (θi ) =

0

r sin θi +r(θi ) cos θi [r(θi ) cos θi + x0 ] c τ

In the above equations r = dr(θ) . Hence, the denomidθ nator in the above expression assumes the following form: r 2 + r 2 = [1 + h(θi )]2 + 2 h2 . In Eq. (A5) an explicit form of Isynapse is Isynapse


N β B g , = B1 N j =1 {1 + β + exp[−r cos(θj )/2]}

− h(θi ) sin θi ] + O( 2 ).

N β (Vth sin θi − 0.5 sin 2θi ) γ j =1 exp(−0.5 cos θj ) + (Vth sin θi − 0.5 sin 2θi ) 2γ × h(θj ) cos(θj ) − 2h(θi )(Vth sin θi − 0.5 sin 2θi )

j =1

β . {1 + β + exp[−r cos(θj )/2]}

(A10)

The above expression can be rearranged and written in the following manner: S=

N

j =1

+ {h cos2 θi − h(θi ) sin 2θi − Vth [h cos θi − h(θi ) sin θi ]} + O( 2 ). (A15)

(A9)

Also, from Eq. (A7) the mean field of synaptic action S is N

β .

1 + β + exp[−0.5 cos(θj )] exp − 2 h(θj ) cos θj

Equation (A13) describes the modification of the mean field of synaptic action for the population of neurons under a small deviation of the amplitude from circularity, and Eq. (A15) accounts for the corresponding total synaptic current. All derivations are carried out without assuming a specific form of function h(θ ). In the main text we approximated numerically an expression for h(θ ) including the first leading expressions of a Fourier series. Assuming here a simpler specific form such as h(θ ) = sin θ for the illustration of the correction term in the synaptic current, Eq. (A15) can be expressed as

(A11) A Taylor series expansion for small in the denominator of Eq. (A11) yields S(θj ) =

(A14)

Isynapse = g/N

and B1 is [1 + h(θi )]2 + 2 h2 . When carrying out a Taylor expansion of 1/B1 for small , we obtain

S=

(A6)

We rewrite the above equation by collecting O() contributions:

B = (Vth sin θi − 0.5 sin 2θi ) + [h cos2 θi − h(θi ) sin 2θi ] − Vth [h cos θi − h(θi ) sin θi ] + O( 2 ), (A8)

1 = [1 − 2h(θi )] + O( 2 ). B1

for r(θi ) cos θi x0 .

+ [h cos2 θi − h(θi ) sin 2θi ]−Vth [h cos θi

(A7)

where we have collected the various orders of in the terms of B and B1 , in particular,

for r(θi ) cos θi < x0

Isynapse

N 1 β

. exp(−0.5 cos θ ) j γ 1− h(θj ) cos θj + O( 2 ) j =1 2γ

(A12) We can further approximate Eq. (A12) in and obtain N exp(−0.5 cos θj ) β 2 S(θj ) = 1+ h(θj ) cos θj +O( ) , γ 2γ j =1

N β (Vth sin θi − 0.5 sin 2θi ) = g/N γ j =1 exp(−0.5 cos θj ) + (Vth sin θi − 0.5 sin 2θi ) 2γ × sin θj cos θj − 2 sin θi (Vth sin θi − 0.5 sin 2θi ) + [cos3 θi − sin θi sin 2θi − Vth (cos2 θi − sin2 θi )] + O( 2 ).

(A16)

(A13)

As → 0 the total synaptic current in Eq. (A14) reduces to the same expression as obtained in Eq. (24).

where γ = [1 + β + exp(−0.5 cos θj )]. Equations (A7) and (A8) together with Eqs. (A9) and (A13) can now be condensed into a single equation that describes the total synaptic current in the phase model

APPENDIX B: COMPARISON BETWEEN THE COLLECTIVE DYNAMICS OF A NETWORK OF NEURONS AND A NETWORK OF A PHASE MODEL

Isynapse

N exp(−0.5 cos θj ) β 1+ h(θj ) cos θj = g/N γ 2γ j =1 × [1 − 2h(θi )](Vth sin θi − 0.5 sin 2θi )

In this section we test the generality of the couplings at the network level that we derived so far in this work. We computationally test our proposition in this paper by considering a globally synaptically coupled network of Class I excitable neurons such as Hodgkin-Huxley (HH) neurons [44].

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PHASE DESCRIPTION OF SPIKING NEURON NETWORKS . . . I

I

(a)

(b)

1 Synchrony a

Synchrony

Monostable

Monostable

B is ta b le

1


a

n˙i = 5 ani Vi (1 − ni ) − bni Vi ,

h˙i = 5 ahi Vi (1 − hi ) − bhi Vi ,

(B3)

s˙i = asi (Vi )(1 − si ) − si /β,

(B4)

(B2)

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g0

V˙i = −0.1(Vi + 65) − 35m3∞ hi (Vi − 55) − 9n4i (Vi + 90) + I + gS(xi − Vth ), (B1)

where all the gating functions used here are taken from Ref. [45]. As before the mean synaptic action is given 1 by S = N1 N i=1 si , where asi (Vi ) = [1+exp(−Vi /2)] . Here we simulate a network of 100 neurons of the above type with random initial conditions and compare the (I, g) parameter space with a globally coupled network of phase model as in Eq. (24). As appears in Fig. 7 from the partitioning of the parameter space (I, g) to a good degree, the dynamics of an ensemble of biologically realistic neural oscillators resembles the dynamics obtained from phase-coupled neural oscillators. It is noteworthy there are some differences in the dynamics of these two networks that arise as we take I close to 0.0. In case of HH networks we do not obtain any further states in the parameter space; however, for a network of phase-coupled neurons we obtain a bistable behavior in a narrow region of parameter space.

0 Incoherence

Incoherence

-1

-1 0 0.2

0.4

0.6

I

0.8 1

0 0.2

0.4

0.6

0.8 1

I

FIG. 7. Parameter space with global synaptic coupling for (a) phase model and (b) HH model.

A single neuron is described by standard HH equations, and synaptically coupled network equations are

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