Multiple-spike waves in a one-dimensional integrate-and-fire neural network

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1 J. Math. Biol. 48, Mathematical Bioloy Diital Object Identifier DOI: 1.17/s Remus Oşan Rodica Curtu Jonathan Rubin Bard Ermentrout Multiple-spike waves in a one-dimensional interate-and-fire neural network Received: 7 Auust 22 / Revised version: 15 April 23 / Published online: 2 Auust 23 c Spriner-Verla 23 Abstract. This paper builds on the past study of sinle-spike waves in one-dimensional interate-and-fire networks to provide a framework for the study of waves with arbitrary finite or countably infinite collections of spike times. Based on this framework, we prove an existence theorem for sinle-spike travelin waves, and we combine analysis and numerics to study two-spike travelin waves, periodic travelin waves, and eneral infinite spike trains. For a fixed wave speed, finite-spike waves, periodic waves, and other infinite-spike waves may all occur, and we discuss the relationships amon them. We also relate the waves considered analytically to waves enerated in numerical simulations by the transient application of localized excitation. 1. Introduction Travelin waves in networks of neurons with purely excitatory synaptic couplin have been the object of many recent theoretical studies [2,3,7,8,1,11,14 18]. These studies are motivated by experiments in which a slice of cortical tissue, with all inhibition blocked, is subjected to a local shock stimulus. This stimulus results in a wave of activity propaatin across the network [4,5,9,13,22]. Theoretical models of this phenomenon rane from continuum firin rate models [1, 18] to simplified spikin models [2,7,17] to detailed conductance-based models [9,22]. Firin rate models do not include individual spikes; as a result, the temporal details of neuronal activity cannot be considered. In spikin models and in experiments, it becomes apparent that after the first wave front passes throuh a network, a sinle neuron can fire many times [9, 19]. However, theoretical analysis of spikin models has, with few exceptions, required that each neuron fire exactly once durin wave propaation. This is either a priori imposed on the model or implemented by stron synaptic depression or after-hyperpolarization. In the sinle-spike case, the existence of travelin waves is reduced to solvin a certain nonautonomous boundary value problem [7, 17]. This computation can be done explicitly when the individual neurons are modeled by the leaky interate-and-fire LIF model. Ermentrout [7], Bressloff [2] and Golomb and Ermentrout [1, 11] developed R. Oşan, R. Curtu, J. Rubin, B. Ermentrout: Department of Mathematics, University of Pittsburh, Pittsburh, PA 1526 USA. Correspondin author: rubin@math.pitt.edu Key words or phrases: Travelin waves Interate-and-fire network Excitatory synaptic couplin

2 244 R. Oşan et al. methods for studyin the existence of travelin waves of activity in networks of LIF cells, incorporatin a variety of additional features such as synaptic delays, aain under the assumption that each cell only fires once. Under this assumption it is also possible to obtain an expression for the wave velocity c [16]. In this paper, we aim to address several questions related to networks of spikin neurons in which each cell fires multiple spikes durin wave propaation. As with most previous analysis, we will restrict our attention to an excitatorily coupled network of LIF neurons. Recall that the LIF model for an individual neuron has the form dv τ 1 = V It, dt where Itrepresents inputs and τ 1 is the membrane time constant. If Vt = V T, the voltae threshold, then Vt = V R, the reset voltae, and the neuron emits a spike. We can formally rewrite this equation to take into account the resettin as dv τ 1 = V It Ṽ R δt t n 1 dt where Ṽ R = τ 1 V R V T and t n denotes the sequence of firin times of the neuron; that is, Vtn = V T for each n 1. Henceforth, we omit the minus superscript when takin the limit of V as t approaches a firin time from the left. We consider a continuous network of such neurons, coupled in a translationally invariant manner on an infinite one-dimensional domain and parameterized by the spatial variable, x. The model is identical to those studied in many of the papers mentioned above. Couplin between a neuron at position x and one at position y is mediated by a time-dependent current with maximal strenth dependin on the distance, x y. Each time a neuron fires, it activates a potential defined by a fixed function, αt, which vanishes for t<, is typically positive for t>, and decays to zero as t>increases. With these considerations, the network of interest is: V τ 1 = Vx,t dy Jx yαt tn t y n= n= n δt t n x Ṽ R 2 for x, t R R, where Ṽ R is iven in 1; we assume that V R <V T. In this continuous network, note that the firin times tn x have a spatial dependence. Interation of 2 yields Vx,tn x = V T and Vx,tn x = V R, which verifies that the constant Ṽ R is defined appropriately. In 2, the parameter denotes maximal synaptic couplin strenth, while Jx : R R {} is the synaptic couplin function, with interal 1. Any interable, even, non-neative function could be used here. In this paper, we take { αt = e t/, t < Ht = e t/τ 3 2, t

3 Interate-and-fire neural network 245 where Htis the Heaviside step function, τ 1 < and Jx = 1 2 e x /. 4 Our results will extend to any qualitatively similar pair of functions; however, these specific choices allow us to make explicit calculations which would become unwieldy in more eneral cases. Other examples are discussed in [14]. Numerical simulations indicate that the type of neural network iven in 2 can support a reat variety of constant speed travelin wave solutions in which each cell fires multiple spikes. Fiure 1 shows an example of travelin waves emanatin from a transient localized stimulus. Bressloff [3] derives a dispersion relation between wave speed c and wave number k in the special case of periodic travelin waves, with tn x = kx nt for each inteer n and c = kt 1. He also characterizes how this relation depends on rise-time of synaptic couplin, synaptic delays, and dendritic cable properties. In this paper, we provide a framework for the study of travelin waves with arbitrary finite or countably infinite collections of spike times. We first, in Section 2, provide a reformulation of 2 which is particularly suitable for the study of travelin waves. In Section 3, we use this to provide a necessary and sufficient condition for the existence of sinle-spike travelin waves, thereby completin the partial study of such waves beun in [2,7], and to analyze two-spike travelin waves k = 5 k = 4 k = 3 k = 2 k = 1 k = 1 8 t * k x x Fi. 1. Numerically simulated travelin waves. Time increases alon the vertical axis, while space increases alon the horizontal axis. The curves shown indicate the times and positions at which spikes occur. At the start of the simulation, all cells are at rest, and cells in the leftmost reion shown are iven a transient excitatory input. No further inputs are iven; all subsequent waves emere spontaneously. The simulation is performed on a spatial reion that is symmetric about x = ; symmetric waves propaate out to the left neative xinthe simulation but are not shown here.

4 246 R. Oşan et al. Ermentrout [7] showed that for fixed parameters, wave speed depends only weakly on the number of spikes fired by each cell in an interate-and-fire network. Oşan and Ermentrout [15] showed similar results for a network of theta neurons another simple one-variable model for neurons. In numerical computations, wave propaation can be initiated by application of a transient excitatory stimulus at a fixed time and location. In these simulations, as waves propaate throuh the network, it appears that the interspike intervals for cells with lare x become independent of x. We analytically calculate the interspike interval for a two-spike wave here, and our results yield an excellent areement with the interspike interval at lare x for multiple-spike waves simulated numerically. Movin beyond finite-spike solutions, in Section 4 we consider travelin wave solutions for which each cell spikes at an infinite sequence {T n x},n, of spike times. Our travelin wave formulation can be used naturally for the iterative computation of the interspike intervals T n1 T n that must arise for such a solution to be consistent with equations 2, 3, 4. Next, we analyze periodic solutions, derivin a three-branched dispersion relation between wave speed c and period T. Finally, we ive some indications that in certain parameter reimes, the interspike intervals of infinite-spike travelin waves with speed c monotonically decrease towards the period T for the periodic solution with the same speed, as n. 2. Travelin wave description We bein by considerin constant speed travelin wave solutions of 2 for which each cell has a finite first spike time. That is, we exclude periodic solutions, which we discuss in Section 4.2. For non-periodic solutions, the n-th spike time, n, of the neuron at the position x can be written as t n x = x c T n. Here we assume T =, and {T n } n is a sequence of nonneative numbers T = <T 1... T N..., with strict inequality as lon as the T n are finite. Travelin wave solutions of 2 take the form Vx,t= Vξfor travelin wave coordinate ξ = tc x R, where c denotes the travelin wave velocity. In terms of this coordinate, and under the assumption that each cell s first spike occurs at a finite time, equation 2 becomes τ 1 cv ξ = Vξ n= duju ξαu/c T n δξ/c T n Ṽ R. 5 n= A travelin wave solution of 5 is obtained by direct interation. For such a solution to be valid, it must satisfy a self-consistency condition, which we state here. This consistency condition relates the asymptotic behavior of V as ξ with the fact that V reaches threshold for the first time at ξ =. In the limit as ξ, the synaptic input to each cell becomes that is, all wave fronts become infinitely far away from each cell. Since solutions of the equation τ 1 cv ξ = Vξdecay to, the consistency condition states that the potential of each neuron must satisfy lim ξ Vξ =. Upon interation of 5 from ξ =to ξ =, usin

5 Interate-and-fire neural network 247 V = V T, this condition formally yields V T = 2 τ 1 c n= e ctn 1 1 c. 6 The derivation of the consistency condition 6 will become more clear in subsection 2.2. For expression 6 to be meaninful, a second condition must hold, namely that the series n= e ct n/ is converent; we will only consider travelin wave solutions for which this is true. Clearly this holds if each neuron fires only a finite number M of times. In this case, we obtain T = <T 1 <...<T M 1 < and set T n = for all n M; thus, the series becomes the finite sum n= e ct n/ = M 1 n= e ct n/. Usin the consistency condition 6, interation of 5 up to arbitrary ξ yields Vξ= V T e ξ/τ 1c I syn ξ Rξ. 7 The function Rξ is the decayin reset, encodin the refractoriness of a cell after a spike. This is iven by Rξ = n= ηξ/c T n where {, t ηt = V R V T e t/τ 1, t > 8 and I syn ξ is the synaptic interal I syn ξ = [ ξ τ 1 c e ξ/τ 1c ds n= ξ e ξ/τ1c /τ 1 c ds n= ] duju ct n sαu/c e s/τ 1c I n s e s/τ1c. 9 On each interval between two consecutive spikes the decayin reset has the form Rξ = V R V T, ξ N 1 n= et n/τ 1 e ξ/τ1c, ct N 1 <ξ ct N N 1. The balance between the input from the synaptic interal and the reset after spikin determines what types of constant speed wave fronts can propaate in the neural network.

6 248 R. Oşan et al Computation of synaptic currents We now derive the synaptic current due to the n-th front of the travelin wave, I n s = duju ct n sαu/c, at some point s on the travelin wave coordinate axis. Suppose we freeze the time t and record what happens at each position in space alon the neural network. Without loss of enerality, we fix our point of reference at x =, where by assumption the first spike occurs at t = such that ξ = ct x =. For any fixed neative time t, none of the wave fronts has yet reached the point x =, and all synaptic current results from waves that will arrive in the future. The n-th front n will reach x = at time T n. Hence, one can derive the position y n of the n-th front at time t<t n from y n = ct n t, which ives y n = ct T n <. Correspondinly, the current measured at x = that is induced by this front future wave at time t is I n;f t = ct Tn dy Jy αt y/c T n. Written in the wave coordinates s = ct x = ct, u = ct y ct n = s y ct n, and usin the fact that J is an even function and that u ct n s = y>, this becomes I n s I n;f s = = 2 duju ct n s αu/c due uctn s/ e u/τ2c = e ct n/ 21 τ 2 c es/. In summary, for t and thus s neative, all the synaptic currents correspond to future waves I n; f and the total current at s is I total s = I n s = n= e ct n/ 21 n= c e s/. 1 At any fixed nonneative time t such that s = ct, say between two consecutive spike-times T N 1 <t T N, there are previous wave fronts that have already passed throuh x = and many others that have yet to arrive. The position reached by each front at the moment t can be found by the same formula, y n = ct T n, as above; the only difference is that y n > for all previous waves n =,...,N 1 and y n for all future waves n N. This classification of waves is illustrated in Fiure 2. The synaptic currents are characterized below: Synaptic current due to a future wave n N I n s = I n; f s = = ct Tn dy Jy αt y/c T n duju ct n s αu/c = e ct n/ 21 τ 2 c es/.

7 Interate-and-fire neural network 249 t T 2 f p p x Fi. 2. Illustration of incomin waves relative to the cell at x = at the time labelled with the solid black circle on the t-axis. At that time, the two waves labelled with f are future waves for this cell, as they have not yet reached x = ; one of them will arrive at time T 2 and the other at time T 3 not shown. The two waves to the riht of the diaonal dashed line are previous waves, as they have already passed throuh x =. We subdivide the synaptic contribution from these waves into p and p, below and above the horizontal dashed line, respectively; these correspond to synaptic inputs from spikes that occurred for some t< and from spikes that occurred for some time t>, respectively. Synaptic current due to a previous wave n =,...,N 1 I n s = I n; p s = where and I n; p s = ct Tn dy Jy αt y/c T n = I n; p s I n; p s dy Jy αt y/c T n = duju ct n s αu/c s ct n = due uctn s/ e u/τ2c = 2 s ct n I n; p s = = ct Tn s ctn dy Jy αt y/c T n duju ct n s αu/c etn/τ2 21 c e s/c.

8 25 R. Oşan et al. Total current = s ctn due uctn s/ e u/c 2 = et n/ 21 τ 2 c e s/τ2c ectn/ 21 τ 2 c e s/. I total s = I n s = n= = N 1 n= N 1 n= et n/ c2 I n; p s I n; f s n=n e s/τ2c N 1 n= ect n/ 21 c e s/ n=n e ct n/ 21 τ 2 c e s/ The travelin wave solution Once the synaptic interal I syn ξ in 9 is computed by interatin I total s = n= I n s, the riht hand side of expression 7 for the solution Vξ is completely specified. The necessary condition 6 that we imposed at the beinnin of our analysis now appears in the form of I syn s for ξ, i.e. I syn ξ = n= e ct n/ 2 τ 1c e 11 τ ξ/ 2 c e ξ/τ 1c. That is, if 6 holds, then the terms V T e ξ/τ1c and I syn ξ in 7 sum to V T e ξ/, such that Vξ asξ. Moreover, as we expected, the equations V = V T and VcT N = lim ξ ct N Vξ= V R V T VcT N = V R are valid. These results are summarized in Lemma 1 below. A more concise expression for Vξis provided in Theorem 1; however, we shall see that for practical purposes, Lemma 1 is very useful. Lemma 1. If condition 6 is true, then the followin function Vξ, ξ = tc x,is a travelin wave solution of the intero-differential equation 2, if all of the terms convere as ξ,n. Vξ= V T e ξ/, ξ, Vξ= [ N 1 V T n= e ctn/ 2 τ 1 c 11 c N 1 n= etn/ 1 2 τ τ 1 c2 V R V T ] τ2 e ξ/c N 1 e ξ/ n= ectn/ 2 τ 1 c 11 c e ξ/ N 1 n= etn/τ τ τ 1 c2 τ2 e ξ/τ 1c N 1 n= et n/τ 1 e ξ/τ1c, ct N 1 <ξ ct N N 1. 12

9 Interate-and-fire neural network 251 Theorem 1. [eneral travelin wave solution] If condition 6 is true, then the followin expression for Vξ, ξ = tc x, denotes a travelin solution of the interate-and-fire model 2, if it converes: Vξ= ηξ/c T n n= n= Ju ξ ct n Au/c du. 13 In 13, η is defined by 8 and A is defined as the convolution function At = α βt= t αs βt sds with βt = τ 1 1 e t/τ 1, i.e., t At = 1 1 τ 1 / e t/ e t/τ. 1, t > Remark 1. For any travelin wave solution with a finite number of spikes, as discussed in the next section, converence is not an issue. 3. Solutions with a finite number of spikes 3.1. One-spike travelin waves We focus first on the case of a solitary wave with speed c and correspondin firin time t x = x/c. In the notation introduced in Section 2, T = and T N = for all N 1. Therefore equation 6 reads V T = 2 τ 1 c τ 2 c and can be solved exactly for c,if/v T 2 1 τ1 2. This necessary condition for the existence of a one-spike wave was used as an existence criterion in [3,7, 16]. When this condition holds, there exist two candidate solutions, the slow wave and the fast wave, correspondin to c slow ; fast = 2τ 1 τ 1 1 τ τ 1 2V T 2V T As /V T increases from its minimal critical value to infinity, c slow decreases from / τ 1 to zero and c fast increases from / τ 1 to infinity. We will denote the 2 curve /V T = 2 1 τ1 as 1F below. In what follows, we analyze the fast one-spike travelin wave, since only this one is stable [2,7]. We will simply write c for the velocity c fast. If a travelin wave solution to 2 exists, then it takes the form Vξ= V T e ξ/ when ξ. When ξ>it is iven by, from Lemma 1,.

10 252 R. Oşan et al. Vξ= V R V T e ξ/τ 1c e ξ/c 2 τ 1c c2 1 τ 1 e ξ/ 11 c e ξ/τ1c c2 1 τ 1. Note that lim ξ ± Vξ=, V = V T, and V = V R. The next step is to check that the candidate solution above indeed has no other spike after it passes ξ = ; that is, the spikin threshold is never reached aain. Therefore we must verify that Vξ<V T for all positive ξ. This is not true in eneral for reasons that are simple to understand. At fixed V R,as/V T,it becomes increasinly easier for the individual neurons to re-excite and spike aain. We compute in the followin the equation of a curve, call it 1 S, which separates the /V T, V R /V T plane to the riht of 1 F into two disjoint reions: a reion where the one-spike fast wave solution exists and a reion where it does not. Theorem 2. [a necessary and sufficient condition for the existence of the fast one-spike travelin wave solution] The interate-and-fire model 2 has a onespike fast wave solution if and only if 2 τ1 /V T and the reset voltae value satisfies V R /V T >Hy 1, 16 where H is defined by Hy = 1 y y /τ 1 /V T 1c 1 τ 1 y/τ2c y 1 2 τ2 2c2 y /τ1c y τ1 2c2 and y is the unique solution in the interval, 1 of the equation Gy = with Gy = y 2c c y/τ2c c τ 1 c c. 18 When 15 holds, for values of V R for which 16 fails, there exists a positive ξ where the threshold V T is reached aain, so the one-spike condition is violated. Proof. We sketch the proof here and provide technical details in Appendix A. Set y = e ξ/. The condition Vξ<V T for all positive ξ reads as Vy<V T for all y, 1, which is equivalent to Hy< V R /V T 1 with H defined by 17. We analyze Hyand obtain that H y = 2τ 1 c V T y 1/τ1c Gy. The sin of H y is the sin of Gy since for the fast wave we have c / τ 1 > /.

11 Interate-and-fire neural network 253 c On the interval [, 1], the function G satisfies G = τ 1 c c >, G1 = τ 1c c τ 1 c c < ; moreover it can be proved that G has exactly one zero in this interval, say y. Hence, Hy has a maximum at y = y, and further, lim y Hy =and H1 =. Toether, these imply Hy >. In summary, Vξ<V T for all ξ>if and only if V R /V T >Hy 1. Remark 2. A special case occurs at /V T = 41 τ 1 /, where the fast wave has the velocity c = /τ 1 and Vξ= V T 2τ 1 τ 1 V R V T e ξ/. ξ e ξ/ 4 2 e τ 1ξ/ 3 τ τ 1 τ1 2 e ξ/ τ 1 2 In this case, the definition of H chanes in the followin way: the ratio y/τ 1 c y 1 2 /τ1 2c2 = yτ1 c y τ 1 c becomes τ 1c τ 1 c y lny, or equivalently 2 1 y lny. τ 1 c 1 1 τ 1 c 1 In the computation of G, the loarithm cancels and we aain end up with expression 18, i.e. y is aain the unique solution of the equation Gy = on, 1. Remark 3. The curve 1 S which divides the plane /V T, V R /V T into the two reions mentioned above has the equation V R /V T = Hy 1. To see that this really forms a curve in the /V T, V R /V T plane, note that y = y, τ 1,,c, with c dependin on /V T. If we fix all parameters except /V T, then we et Hy = Hy /V T. Remark 4. Recall that the equation /V T = τ1 defines the curve 1 F in the /V T, V R /V T plane. Theorem 2 states that to obtain a one-spike travelin wave in the interate-and-fire model, the parameters must lie to the riht of the curve 1 F for firin to occur, and above the curve 1 S for firin to stop after one spike. That is, the first condition allows for self-sustained propaation of the travelin wave, while the second prevents multiple spikins by resettin the potential to low enouh values. These curves are displayed in Fiure 3 for a representative parameter set.

12 254 R. Oşan et al F EXIST. 1 S 24 V R / V T / V T Fi. 3. The curves 1 F and 1 S for τ 1 = 1, = 2, = 1, V T = 1. Parameter values must lie to the riht of 1 F for cells to be able to fire upon receivin the one-spike synaptic input. Parameter values must lie above 1 S for cells to stop firin after just one spike. Between the two curves, one-spike waves exist in the reion labelled EXIST. Note that 1 S terminates in an intersection with 1 F,at/V T = 21 τ 1 / 2 with V R /V T finite and positive Two-spike travelin waves In a two-spike travelin wave, each cell fires at times that we denote T = and T 1 = T. In our earlier notation, this also means T j = for each j 2. In this case, equation 6 and substitution of the condition VcT = V T into 12 in Lemma 1 read as V T = 1 e ct/ 2 τ 1c 11 τ 2 c, 19 V T = V R V T e T/τ 1 e T/ 2 τ 1c c2 1 τ 1 11 τ 2 c e T/τ1 2 τ 1c e ct/ 11 c c2 1 τ 1. 2 With the definition fc= 2 /V T τ1 c c we obtain accordin to 19 an explicit equation for T, T = ln fc. 22 c

13 Interate-and-fire neural network 255 We are ready now to investiate under which conditions the system 19, 2 has a solution c, T with positive c and T, i.e. when fcis between and 1. Let us define [ ] 2 c 1;2 = 2τ 1 V T τ 1 1 V T τ τ 1, c 1;2 = 2τ 1 [ 2 2V T τ 1 1 2V T τ τ 1 ] 23 and notice that fc= atc = c 1;2 and fc= 1atc = c 1;2. The set S c = {c R fc, 1 } is easily computed: if /V T < τ1 then Sc = ;if 1 τ1 τ </VT 2 < 2 1 τ1 then c1;2 are 2 complex with nonzero imainary parts and S c = c 1, c 2 ;if2 1 τ1 /VT then < c 1 <c 1 c 2 < c 2 and S c = c 1,c 1 c 2, c 2 R. Remark 5. c 1 = c 1 and c slow 2 = c 1 ; that is, c fast 1 and c 2 are the slow and fast velocities from the one-spike wave case. Moreover, Tc as c c 1;2 since we have then fc. Remark 6. As /V T we obtain c 1,c 1, c 2, c 2 and c 2 /c 2 2, c 1 /c 1 1/2. The equations 2 and 22 imply Fc =, where V R 1 fc τ 1 c 1 /V T V T 1 τ 1 Fc = with τ2 c c f c fc τ i c 1 1 τ f τi c = i c 1, c /τ i i = 1, 2. ln f τ i, c = /τ i τ 1c τ 1 c f τ 1 c, The velocities of candidate two-spike travelin wave solutions are precisely the roots of F that belon to the set S c. Such velocities correspond to true two-spike travelin wave solutions if Vξ<V T for all ξ>ct. Lemma 2. The function F : S c R defined by 24 is continuous on S c and satisfies F c 1 = F c 2 = V R V T 1 >. Proof. The result comes directly from the definition of F and the fact that lim c c1;2 fc= 1. Here and below we use the notation Fx = lim x x Fx, Fx = lim x x Fx. Lemma 3. Suppose that /V T 2 1 τ1 2, i.e. Sc = c 1,c 1 c 2, c 2. Then Fc 2 =and Fc 1 <. Moreover 24

14 256 R. Oşan et al. 2 i if 2 1 τ1 /VT < 4 1 τ 1, then c 1 < τ 2 <c 1 c 2 < τ 1 < c 2 and Fc1 =, ii if /V T = 4 1 τ 1, then c 1 <c 1 = τ 2 < τ 1 = c 2 < c 2 and Fc1 =, iii if /V T > 4 1 τ 1, then c 1 <c 1 < τ 2 V T 1 τ 1 2 τ τ1 2c c2 1 τ2 2c2 1 2 <. < τ 1 <c 2 < c 2 and Fc1 = Proof. See Appendix B. These two lemmas immediately imply the followin result. The relation between velocities described in the theorem is shown in the numerical results in Fiure 4. Theorem 3. If /V T 2 1 τ1 2, then for all VR R, there exist two distinct positive values c S c 1,c 1, c F c 2, c 2 such that Fc S = Fc F =. Therefore there exist two distinct solutions c S,T S, c F,T F for the system 19, 2. When these correspond to two-spike travelin wave solutions, the velocity c F c S of the fast slow two-spike travelin wave solution is reater less than that of the fast slow one-spike travelin wave solution. Remark 7. The above results apply for /V T 21 τ 1 / 2. We also expect two-spike travelin waves to exist for some /V T < 21 τ 1 / 2, since S c 6 4 sinle spike double spike velocity Fi. 4. Numerically enerated curves showin wave speed as a function of couplin strenth for one- and two-spike waves.

15 Interate-and-fire neural network 257 for 1 τ 1 / 2 </V T < 21 τ 1 / 2. In fact, in analoy to the one-spike case, we expect that there exist curves 2 F, iven by /V T = F 2 V R /V T, and 2 S, iven by V R /V T = S 2 /V T,inthe/V T, V R /V T plane, such that for all /V T to the riht of 2 F, cells can fire two spikes, while for all V R /V T above 2 S, cells fire at most two spikes. It is perhaps non-intuitive that, for fixed V R and V T, a two-spike travelin wave should exist for smaller than needed for a one-spike wave. This holds because in a two-spike wave, the two spikes fired by each cell produce a larer overall synaptic input to each cell in the medium, which promotes firin. We can carry these ideas further to make several reasoned conjectures. Recall that a cell s voltae is reset to V R after a spike. For fixed V T,as V R increases, a larer is required to elicit a subsequent second spike. Hence, we expect F 2 to have a positive slope, with F 2 V R /V T 21 τ 1 / 2 as V R. For fixed and V T, a sufficiently lare value of V R sufficiently stron reset is required to prevent subsequent spikes after a second one, with a stroner reset needed for larer. Hence, we also expect S 2 to have a positive slope. Finally, the same aruments should ive correspondin curves for N-spike waves, for any positive inteer N, such that N F moves leftwards and N S moves upwards in the /V T, V R /V T plane, as N increases. Fiure 5 illustrates a numerically enerated version of the curve 2 F, the shape of which arees with our conjectures. The expected relation of the curves for one- and two-spike waves is drawn in Fiure 6. The proofs of these conjectures remain open. 8 6 V R /V T /V T Fi. 5. Numerically enerated 2 F curve. To the riht of this in parameter space, cells can propaate two-spike waves.

16 258 R. Oşan et al. V R / VT 2F 1F 1 & 2 2S 2 1 1S / V T Fi. 6. Schematic illustration of the expected relation of the 1 F, 1 S, 2 F and 2 S curves in parameter space. In the reions labelled 1 or 2, one-spike or two-spike waves exist; in the reion labelled 1&2, these co-exist. Outside of the labelled reions, neither type of wave exists. By solvin the equation Fc = numerically for fixed parameters, one can analytically find the velocity c. Given this, equation 22 yields the time T between the two spikes in the correspondin travelin wave if it really is a two-spike solution. These results match quite closely to those obtained from numerical simulation of fast two-spike travelin waves. Numerically, waves are initiated by applyin an excitatory input, or shock, to an initial reion, which leads to wave propaation away from the reion [15,16]; see Fiure 1. Note that wave solutions obtained in this manner are conceptually different from those studied analytically, in that the numerical waves oriinate at a finite time, from a specific spatial location. Thus, it is rather interestin that wave speed c and second spike time T from theory and numerics compare so well. 4. Arbitrary numbers of spikes and infinite spike trains 4.1. Computation of interspike intervals Consider a travelin wave solution for which each cell spikes at an infinite sequence {x/c T n }, n, of spike times. We will discuss here how the formulation iven in Lemma 1 can be used to compute the interspike intervals T n T n 1 between successive waves. In the travelin wave formulation, in which V is expressed as a function of ξ = ct x, wehavevct n = V T for each T n. Correspondinly,

17 Interate-and-fire neural network 259 Lemma 1 implies that for any N 1, N 1 n= V T = e ct n/ V T 2 τ 1c 11 τ 2 c e ct N / N 1 n= ect n/ 2 τ 1c 11 c e ct N / V R V T c2 1 τ 1 N 1 n= et n/ 1 2 τ c2 N 1 e T n/τ 1 e T N /τ n= e TN /τ2 Suppose that there are a finite number of spikes in the travelin wave, say N 1. For each spike, equation 25 applies. In fact, one obtains a system with N 1 equations and N 1 unknowns to be solved to obtain a valid N 1-spike travelin wave solution. The unknowns are c, which denotes the velocity of the travelin wave, and the spike-times T 1 up to T N since T = by convention. The equations in the system are those correspondin to VcT n = V T, 1 n N, and the equation 6. Based on the analysis in the previous section, it appears that this hihly nonlinear system can only be solved numerically for most N. The situation becomes even more complicated when an infinite number of spikes is considered. Thanks to the travelin wave description set out in Section 2, equation 25 is available, and one can iteratively solve for the spike time T N from the previously known spike times T =,T 1,..., T N 1 for every N 1. To do so, however, an obstacle must first be overcome: since equation 6 involves all of the travelin fronts, it cannot be used independently from 25 in the case of infinitely many spikes. Thus, the velocity c that appears in each equation must be determined from some alternate source and then used here as a constant. Once c is specified from such a source, one can iteratively compute the spike times, and hence the interspike intervals T N = T N T N 1. Remark 8. One source for the wave speed here is the fast two-spike wave speed calculated from the analytical formulas 19, 2. Numerics show that the speeds of waves with lare numbers of spikes are quite similar to those calculated for two-spike waves with correspondin parameter sets. Intuitively, this makes sense because in fast two-spike waves, the interspike interval ct is sinificantly reater than, the space constant or footprint of the synaptic couplin; see Fiures 3, 5. Even for ct = 2,wehaveJcT = 1/2e 2, such that little interaction occurs between the synaptic inputs from different waves in the same solution. Thus, waves travel with rouhly the same speed, no matter how many waves there are. One miht question the value of computin the interspike intervals from numerical solution of equation 25, iven that one can perform a numerical simulation of travelin waves in the full network. However, such simulations are based on applyin a localized shock somewhere in the network at a fixed time and allowin waves to propaate thereafter [15, 16]. This corresponds to a different, althouh

18 26 R. Oşan et al. closely related, form of travelin wave from that which we analyze analytically, for which all waves can be thouht of as havin existed somewhere in the infinite network for all time. In fact, one interestin result that arises from usin equation 25 to compute interspike intervals is that we can compare theory and analysis, to see just how closely related these forms of travelin wave solutions are. To make this comparison, we used the parameters τ 1 = 1, = 2, = 1, V T = 1, V R = 25 and = 6. For the first six interspike intervals, full numerical simulation of equation 2, labelled as Numerics, and numerical solution of the analytical expression 25, labelled as Iterations, produced excellent areement, as shown in the followin table. Note that for the analytical approach, we used c = , the speed of the wave found in our numerical simulations. Numerics I terations Error T 1 T e 6 T 2 T e 5 T 3 T e 5 T 4 T e 4 T 5 T Errors in the table row due to difficulty in solvin equation 25 numerically, resultin from the fact that the first product in this equation consists of a factor that converes to as N with a factor that diveres as N. Thus, we stopped after computin six interspike intervals. In full numerical simulations, the subsequent interspike intervals can be computed as well. We did this at a distance correspondin to about 4 away from the oriinally shocked reion of about 5, where is the footprint of the couplin function Jxin 4. The total lenth of the domain is 1 so that a point 4 from the center is still far enouh away from the boundary to avoid any ede effects. We note that these intervals form a monotone decreasin sequence, which appears to convere to an asymptotic value. This will be relevant when we discuss periodic solutions in Section 4.2. Remark 9. It can be shown, by induction on N 1, that equation 25 is equivalent to V T = V T N 1 n= e ct n/ 2 τ 1c 11 c N 1 n= ect n/ 2 τ 1c 11 τ 2 c e ct N / N 1 n= et n/ c2 1 τ 1 e TN /τ2 e ct N / [ 1 e T N T N 1 1 τ 1 c ] [ 1 e T N T N 1 1 τ 1 c ] [ ] 1 e T N T N 1 τ τ 2 V R e T N T N 1 /τ This formulation proves its advantae when we discuss the effect of addin an absolute refractory period to the interate-and-fire model Section 4.4.

19 Interate-and-fire neural network Periodic solutions We next analyze periodic travelin wave solutions, for which the time interval between each pair of spikes fired by any fixed neuron is a constant T>. Bressloff offers a eneral account of this case, but he presents his results in terms of infinite Fourier series expansions [3]. We use the formalism developed in Section 2 to obtain the dispersion relation between velocity c and period T directly. A periodic travelin wave solutions exists precisely for those values c, T that satisfy the dispersion relation. Our explicit calculations allow us to plot the dispersion relation, which ives insiht into the rane of speeds and periods for which periodic solutions can exist. We assume that all travelin wave fronts have the same velocity c, such that they are separated by a distance ct at all times. The case of a periodic solution is different from all others we have treated, because for fixed x, we can no loner identify a first spike time x/c T. Equation 2 must be modified accordinly, takin into account the fact that at each point in space, when we record a new spike, we assume that an infinite number of fronts have already passed and an infinite number will come. This means that the spike times should be defined as t n x = x/c T n = x/c nt with n {..., 2, 1,, 1, 2,...} and the sums in the synaptic interal and the reset term should be taken over all inteers. Under this ansatz, we analytically derive the dispersion relation, which consists of curves relatin wave velocity c and time period T. To understand why periodic solutions exist for intervals of c and T values, rather than for a unique pair c, T, note that an equation similar to 6 does not arise in the periodic case, since the assumption lim ξ Vξ= does not apply. Suppose that a cell spikes at times and T durin a periodic solution. The two mathematical conditions that encode these spikes are V = V R and VT= V T, but, as in the previous sections, these are redundant. Therefore, the two unknowns c and T are related by only one equation Derivation of the dispersion relation Aain, we fix a point x = as the oriin for our reference frame and record, at arbitrary fixed t,t], the strenths of the synaptic inputs that reach x = from the inteer-indexed sequence of travelin wave fronts in the solution. These inputs are classified as previously: Synaptic current due to future waves Here we consider wave fronts that are at some positions y n at time t and will reach x = in a time equal to nt, for n 1. That is, y n = ct nt and tn x = x c nt, for each fixed x R. The correspondin synaptic current is iven by I n; f t = = 2 = ct nt ct nt dy Jy αt y/c nt dy e y/ e t y/c nt / e nct / 21 c ect/, n 1. 27

20 262 R. Oşan et al. Synaptic current due to previous waves Here we consider wave fronts that are at positions y n > at time t. These fronts pass throuh x = at times equal to nt for n, such that y n = ct nt and tn x = x c nt for each x for this front. The correspondin synaptic current is iven by where and I n; p t = I n; p t = = 2 = ctnt dy Jy αt y/c nt = I n; p t I n; p t I n; p t = ctnt ctnt 21 c = 2 dy Jy αt y/c nt dy e y/ e t y/cnt / = e nt / 21 c e t/, n 28 dy Jy αt y/c nt dy e y/ e t y/cnt / e nt / e t/ e nct / e ct/, n. 29 Total synaptic current The equations 27, 28 and 29 sum to ive the total synaptic current I total t = I n; f t I n; p t I n; p t n=1 n= e ct/ e t/ = 21 τ 2 c e ct / 1 21 τ 2 c 1 e T/ e t/ 21 τ 2 c 1 e T/τ e ct/ 2 1 e ct / e t/ e ct/ = e T/τ 2 21 τ2 2c2 c 1 e ct / e ct/ 21 τ 2 c e ct /. 3 1 By substitutin 3 into the modified form of equation 2 and interatin over the time t,t], usin the condition V = V R or equivalently VT = V T, we obtain the dispersion relation Vc,T= V T 31 n=

21 Interate-and-fire neural network 263 where Vc,T=V R e T/τ 1 e T/ e T/τ τ c2 1 e T/ e ct / e T/τ 1 2 τ 1c 11 τ 2 c 1 e ct / e ct / e T/τ 1 2 τ 1c 11 τ 2 c e ct / Numerical computation of the dispersion relation It is a simple matter to solve 31 numerically for a iven set of parameters and thus find c as a function of T.We plot Vc,Tfrom 32 for fixed c and = 6, = 1, τ 1 = 1, = 2, V T = 1, V R = 25 in Fiure 7. The full dispersion relation for these parameters is shown at the left of Fiure 8. Note that there are three branches of solutions to 31. This represents a sinificant difference from the dispersion relation for periodic solutions of the FitzHuh-Naumo reaction-diffusion model for nerve conduction, which features two connected branches of solutions c f T and c s T, joined at a minimal T value [2]. The riht part of the fiure shows a different view of the dispersion relation, formed by plottin the temporal frequency, ω = 1/T, as a function of the wavenumber, k = 1/cT. Heuristically, we can understand the three-branch structure of the dispersion relation in the followin way. For fixed c,ift were increased from 1.6, each cell in the network would receive less and less current at each fixed time. Correspondinly, Vc,T T Fi. 7. Vc,T versus T for the parameter values iven in the text, with fixed c Points on the dispersion relation are iven by intersections of Vc,Twith V T, which here has been chosen to equal 1.

22 264 R. Oşan et al. 1 c T ω k Fi. 8. The dispersion relation for = 6, = 1, τ 1 = 1, = 2, V T = 1, and V R = 25. Left fiure: Velocity c versus period T. The left branch oriinates with T and has a vertical asymptote at T The upper riht branch has a vertical asymptote at T and a horizontal asymptote at c = 1. The lower riht branch oriinates with T and has a horizontal asymptote at c = 1/2. No periodic solutions exist with periods between the two vertical asymptotes. Riht fiure: The same data, plotted as frequency ω = 1/T versus wave number k = 1/cT. c must increase to deliver enouh current to sustain periodic spikin. Thus, the low T branch of the relation is current-limited. On the other hand, for very lare T, spikes essentially do not interact. Thus, it is not surprisin that periodic solutions can occur with approximately the speeds of the fast and slow sinle-spike solutions; once time T elapses after a cell spikes, the cell s voltae has recovered to approximately the riht level to propaate a new sinle-spike wave. Consider the upper branch, correspondin to the fast wave. As T decreases from lare values, if c were kept fixed, each cell would actually receive too much input to maintain a lare T periodic solution. Faster speeds c are required to carry away waves that have already passed a cell, allowin the cell to recover to a state from which it can respond aain. Thus, the lare T, lare c branch of the relation is recovery-limited. Similarly, due to the inverse relationship between speed and period that characterizes slow waves, slower speeds are needed to compensate for decreases in T, yieldin the lare T, small c curve Converence of interspike intervals In addition to the above heuristic interpretations of the three-branched dispersion relation, there is a natural correspondence between two of the three branches of periodic solutions iven in the dispersion relation and our numerical simulations of waves, induced by a localized shock, in the full network. The low T branch of

23 Interate-and-fire neural network 265 periodics corresponds to experiments in which a sinle shock is applied, while for the lare T - lare c branch, spike interactions are limited, such that one would not expect to obtain these solutions in a sinle shock simulation. As an example, consider the raph of Vc,T versus T, with c fixed at approximately 1.25, the wave speed obtained from numerical simulations of the full network. This raph is presented in Fiure 7. Since V T = 1, it is easy to see that the equation Vc,T = V T has solutions at two values of T. The smaller one occurs at T Recall that when we solved the interate-and-fire equation 2 numerically, after applyin a shock to an initial reion, we observed that interspike intervals decayed monotonically. For example, in addition to the interspike intervals presented in the table in subsection 4.1, we find that the tenth throuh sixteenth interspike intervals are , , , , , , and We expect that these interspike intervals convere asymptotically towards the value of T for the low branch periodic solution with the same c Due to the close relation between the analytically derived infinite-spike solutions considered in Section 4.1 and the solutions observed in our numerical simulations, this example suests that two results hold, at least in certain parameter reimes. First, for fixed wave speed c, the interspike intervals of the analytically derived infinite-spike travelin waves convere in a monotone decreasin way to the low branch period of the periodic solution for that c. Second, periodic solutions are stable and are attractors for the solutions obtained in the types of numerical simulations described here. To follow up on the first of these conjectures, for fixed c>, suppose that an infinite-spike travelin wave solution of 2 exists with spike times {x/c T n }, n, for each x, and without loss of enerality set T =. We will derive a condition under which T 1 T T 1 >T, where T is the period of the low branch periodic solution with the same c. Henceforth, we assume cτ 1 >, based on the fact that this inequality holds for c 2, the analytically computed speed of the two-spike wave. Note the similar structures of the equation V T = V R e T 1/τ 1 e T 1 / e T 1/τ 1 e ct 1 / 1 2 τ c2 e T 1/τ 1 2 τ 1c 11 τ 2 c [ ] V T 2 τ 1c 11 τ 2 c e ct1/ e T 1/τ 1, 33 derived from 26 for N = 1 and satisfied by the infinite-spike wave, and the dispersion relation V T = V R e T/τ 1 e T/ e T/τ τ c2 1 e T/

24 266 R. Oşan et al. 2 τ 1c 2 τ 1c c c e ct / e T/τ 1 1 e ct / e ct / e T/τ 1 e ct / 1 satisfied by the periodic solution. To show that T 1 >T for fixed c, it suffices to show that the riht hand side of equation 33, evaluated at T 1 = T, is less than V T. This will imply that at T = T 1, the cell at x = in the infinite-spike wave has not yet reached V T and fired its second spike, such that T 1 must be reater than T. Equivalently, we can show that the riht hand side of equation 33, evaluated at T 1 = T, is less than the riht hand side of equation 34. Of course, we can nelect the first V R terms, since they become identical at T 1 = T. Further, consider the sums in the first set of square brackets in each equation, evaluated at T 1 = T for 33. For both equations, these terms are positive; however, in equation 34, each term in the sum is divided by a factor of the form 1 e µ for a positive number µ. Hence, the sum from equation 34 is larer than that from equation 33. Thus, it suffices to show that the final term on the riht hand side of equation 33, evaluated at T 1 = T, is less than the final term in the riht hand side of equation 34. Due to the common factors shared by these terms, this condition reduces to V T < 2 τ 1c 11 τ 2 c 1 1 e ct / Since the riht hand side of 35 is nonneative, continuous and monotonically decreasin in positive T, and blows up as T, there does exist a unique T such that inequality 35 holds on,t. So, T 1 T >T as lon as T < T, or equivalently, as lon as T < /c lnh/h where H = 2V T τ 1 c/ 11 / c Absolute refractory period After each spike, real neurons are unable to fire aain durin a time period called absolute refractory period. We address here the consequences of addin this feature to the interate-and-fire model. Equation 1 is supplemented by the condition that durin the refractory period, on the interval,t r, the potential of the neuron is held fixed at V T. While the addition of this feature induces extra complexity in the travelin wave solutions, they retain a similar structure. First, the interspike intervals can be computed from the followin modified version of equation 26 N 1 n= V T = e ct n/ [ ] V T 2 τ 1c 11 τ 2 c e ct N / 1 e T N T N 1 t r τ 1 c 1 N 1 n= ect n/ 2 τ 1c 11 c e ct N / [ 1 e T N T N 1 t r 1 τ 1 c ]

25 Interate-and-fire neural network 267 N 1 n= et n/ c2 1 τ 1 e TN /τ2 [ ] 1 e T N T N 1 t r τ τ 2 V R e T N T N 1 t r /τ We outline here the factors that determine this new formulation. We note first that the formula for the synaptic current 3 does not chane, since the refractory effects apply to the potential of the neuron and not to synaptic contributions. Second, since the cell potential is held fixed durin the refractory period, it follows that the interation interval of equation 3 is reduced from ct k 1, ct k to ct k 1 t r, ct k. Numerical simulations for t r =.3 yield travelin waves with speed c = and the first six ISIs of 2.841, 2.517, 2.397, 2.341, 2.314, 2.3. The numerical scheme for computin ISIs can be used to compute the first three terms, 2.841, 2.52, 2.43, before numerical errors become too lare. The reduced efficiency of the ISI scheme is an effect of includin the refractory period. Remember that the first term in equation 36 is the one responsible for numerical instabilities. Addition of the refractory period pushes T N to larer values, increasin the manitude of the factor e ct N /, which drives the numerical scheme towards instabilities faster. Similar structural chanes apply for the dispersion relationship, iven in its oriinal formulation by equations When the absolute refractory period is introduced in the LIF model, equation 32 chanes to Vc,T=V R e T tr τ 1 2 τ 1c 2 τ 1c 1 2 τ c2 11 τ 2 c 11 τ 2 c e ct tr e ct tr T tr e e T tr τ 1 1 e T/ e T tr τ 1 1 e ct / e T tr τ 1 e ct / 1 e ctr e tr e ctr. 37 Numerical simulations of the full model aain suest the existence of a periodic solution, as the ISI intervals decrease monotonically towards a fixed non-zero value. The tenth such ISI has a value of , in areement with the value T = obtained usin equation 37 with c = The most important chane in the shape of the dispersion curves, as a result of includin the refractory period, is that for lare enouh refractory period durations t r, the vertical asymptotes no loner exist and instead the upper branches are united, as indicated in Fiure 9. However, the the existence of horizontal asymptotes is unaffected, as they correspond to solutions that approach slow and fast sinle-spike travelin wave solution. We point out here how the solutions of equations 36, 37 reflect lobal chanes induced by the refractory period on the dynamics of the travelin waves. Addin

26 268 R. Oşan et al c T ω k Fi. 9. The dispersion relation for = 6, = 1, τ 1 = 1, = 2, V T = 1, V R = 25 and t r =.6. Left fiure: Velocity c versus period T. Note that the left branch connects with the upper riht branch. The newly formed branch oriinates with T 2.78, has a peak at T 9.14 and maintains its horizontal asymptote at c = 1. The lower riht branch oriinates with T 11.8 and maintains its horizontal asymptote at c = 1/2. Riht fiure: The same data, plotted as frequency ω = 1/T versus wave number k = 1/cT. the refractory period to the model slows down wave propaation in the network due to two factors. First, holdin the cell at the reset potential prevents the neuron from returnin to its rest state and also limits the speed with which the cell can reach spikin threshold. Second, durin the refractory period, synaptic contributions decay, thus increasin the amount of time required for the neuron to spike aain. A slower wave speed ensures that when a cell emeres from refractoriness, there is still synaptic current available to push it towards threshold. Thus it is not surprisin that, when comparin numerics for the t r = and t r >, the latter, refractory case exhibits both a decrease in the speed of the travelin waves and an increase in the interspike intervals that separates them at least on the small T part of the dispersion curves. Furthermore, the positive correlation between interspike intervals and refractory period also applies to periodic solutions of equation Summary and open problems Travelin waves of activity are observed in slices of cortical tissue under various pharmacoloical manipulations [4, 5, 9, 13, 22]. These have been used both as models for epilepsy and as a way to probe the intracortical circuitry. Two of the macroscopic quantities which can be readily observed are the velocity of propaation and the number of spikes fired durin a dischare see for example, [9], where this is quantified. In previous work, [7,1,11,14,17], we studied how the velocity of travelin waves depends on various parameters by assumin that each cell spiked once.