Noise-Stabilized Long-Distance Synchronization in Populations of ...

Journal of Computational Neuroscience 15, 143–157, 2003 c 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.


Noise-Stabilized Long-Distance Synchronization in Populations of Model Neurons DAVID MCMILLEN Department of Chemistry, University of Toronto at Mississauga, 3359 Mississauga Rd, Mississauga, ON L5L 1C6 Canada [email protected]

NANCY KOPELL Center for BioDynamics and Department of Mathematics, Boston University, 111 Cummington St., Boston, MA 02215, USA Received October 1, 2002; Revised February 19, 2003; Accepted March 11, 2003 Action Editors: Terrence Sejnowski and Christof Koch

Abstract. Rhythmic, synchronous firing of groups of neurons is associated with behaviorally relevant states, and it is thus of interest to understand the mechanisms by which synchronization may be achieved. In hippocampal slice preparations, networks of excitatory and inhibitory neurons have been seen to synchronize when strong stimulation is applied at separated sites between which any coupling must be subject to a significant axonal delay. We extend previous work on synchronization in a model system based on the network architecture of these hippocampal slices. Our new analysis addresses the effects of heterogeneous populations and noisy inputs on the stability of synchronous solutions in the system. We find that, with experimentally motivated constraints on the coupling strength, sufficiently large heterogeneity in the input currents renders synchrony unstable. The addition of noise, however, restores stable near-synchrony. We analytically reduce the high-dimensional biophysical equations for the full population to a simple three-dimensional map, and show that the map’s stability properties correctly predict both the loss of stability and the restabilizing effect of the noise. Keywords: neural synchrony, stability analysis, noise-enhanced 1.

Introduction

Behaviorally relevant states such as arousal, quiet awareness, and exploration are associated with detectible spectral frequency signatures in EEG recordings (Gray et al., 1989; Murthy and Fetz, 1992; Joliot et al., 1994; Bragin et al., 1995; Pulvermuller et al., 1997; Roelfsema et al., 1997; De Pascalis and Ray, 1998; Farmer, 1998; Gomez et al., 1998; Tallon-Baudry et al., 1998; Miltner et al., 1999; Rodriguez et al., 1999; Tallon-Baudry and Bertrand, 1999; Fries et al., 2001). The well-known β and γ rhythms are associated with

precise synchrony in populations of neurons, and it is of interest to study how such synchronization can arise. It is known that cells can synchronize over distances of at least several millimeters, a distance which implies significant conduction delays between sites. We are interested in the question of how groups of neurons can maintain synchrony despite these delays. In particular, we consider a model for the γ frequency rhythm found in neocortical and hippocampal systems during states of sensory stimulation. Traub et al. (1996) and Whittington et al. (1997) noted that synchronization in the presence of delays was correlated with the

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E1

g ei

E2

g

I1

δ ie

c ei

I2

g ii Figure 1. Two local E-I circuits, coupled through distal E-I connections. Local synaptic connections are taken to be instantaneous, while distal connections (dashed lines) experience an axonal delay of δ; throughout this paper, we use δ = 10 ms.

appearance of spike “doublets” in the inhibitory cells of in vitro hippocampal preparations. In doublet firing, the inhibitory cells generate two spikes between the spiking of the excitatory cells, and such doublets appear in slice preparations in the presence of strong excitation at separated sites. Previous work (Ermentrout and Kopell, 1998) analyzed a mechanism for this type of synchronization in a network of two coupled local circuits (see Fig. 1). An excitatory cell (E-cell) coupled to an adjacent inhibitory cell (I-cell) forms a local circuit, and two such circuits are coupled through connections which experience a substantial axonal delay. The previous study showed how the timing of the second spike in the doublet encodes information about the phases of the local circuits in the previous cycle. By reducing the biophysical equations to a one-dimensional map, it was shown that this timing information could be used by the system to synchronize the excitatory cells in the two separated circuits, and the conditions for stable synchrony were elucidated. In this paper, we extend the previous work in two ways: we replace each single cell in the original model by a heterogeneous population; and we consider the effect of noise on the ability of the system to synchronize. We show that, when studying populations, each block of firing may be represented using just two characteristics: the timing of the firing event, and the fraction of cells participating in the firing. Simulations suggest that the most important of these fractions is the proportion of I-cells firing in the second beat of a doublet. That this fraction changes from cycle to cycle is supported by data showing that involvement of individual interneurons in the

doublet spike can change from cycle to cycle, while the cells fire on every cycle for the first spike of the doublet (Whittington et al., 2000). Incorporating the former feature, we obtain a three-dimensional map, and analyze its stability properties. We examine the case where there is incomplete participation of the I-cells in the second firing event of the doublet, and show, via map analysis and simulations, that moving from single cells to a sufficiently heterogeneous population disrupts the stability of the synchronous solution, that the addition of noise (in the form of random stimulation by excitatory and inhibitory Poisson processes) restores the stability of a near-synchronous solution, and that this stabilizing effect is also predicted by analyzing the map. In a two-dimensional noise-heterogeneity space, we find (by numerical simulation) the region in which the noise-stabilization of synchrony occurs. 2. 2.1.

Methods Model Specification

We seek to mimic in simplified, but not minimal, models some broad features of the CA1 region of the hippocampus. The pyramidal cells and the interneurons are modeled by one-compartment cells. Since we are concerned with the effects of long-distance communication, we consider models that have circuits at separated places, rather than distributed networks. Since a main difference between models involving single cells and models based on populations is the possibility of heterogeneity in the latter, we add such heterogeneity by varying the input current, and thus the intrinsic firing rate, of the cells in each population. 2.1.1. Network Architecture: Two Coupled Local Circuits. We consider two local circuits, each circuit consisting of one population of excitatory cells (Ecells) and one population of inhibitory cells (I-cells), connected as shown in Fig. 1. All numerical results reported are for population sizes of N = 100 (for each of the four populations). The connections between populations are taken to be all-to-all: every neuron in the source population influences every neuron in the target population. Local synaptic conductances are labeled as follows: E-I connection, gei ; I-E connection, gie ; I-I connection, gii . We neglect local E-E coupling, since such connections tend to be sparse in the hippocampus. The distal E-I synaptic conductance is denoted cei ; there is no other distal coupling. In this work, we

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keep the conductances fixed at gei = 0.2, gie = 2, gii = 0.3, and cei = 0.1, with the exception of one example in Section 3.2 where we set cei = 0.2 (all in mS/cm2 ). The distal E-I connections are subject to an axonal delay of δ = 10 ms.

cells αi ∈ [(1 − 0.1H ), (1 + 0.1H )], while for inhibitory cells αi ∈ [(1 − 0.5H ), (1 + 0.5H )].

2.1.2. Membrane Model: 2D Hodgkin-Huxley. We employ a two dimensional reduction of the usual 4-D Hodgkin-Huxley equations, obtained by taking the sodium activation to be fast, m = m ∞ (V ), and taking the sodium inactivation to be a function of the potassium activation, h = max(1 − 1.25n, 0) (Traub and Miles, 1991; Ermentrout and Kopell, 1998). The two remaining dynamical equations give the behavior of the membrane voltage, V ,

ds = A(1 − s)S(V ) − Bs, dt

C

dV = I0 + Isyn + g L (VL − V ) + g K n 4 (VK − V ) dt + g N a m 3 h(VN a − V ), (1)

(2)

C is the membrane conductance, I0 is a current regulating the cell’s spontaneous firing rate, Isyn represents all synaptic inputs from other neurons, and the remaining terms reflect voltage-dependent ion channel responses. The sodium activation is given by m = m ∞ (V ) =

am (V ) , am (V ) + bm (V )

(3)

with am (V ) = 0.32(V + 54)/(1 − exp[−(V + 54)/4]) and bm (V ) = 0.28(V + 27)/(−1 + exp[(V + 27)/5]). For the potassium activation, n, the voltage-dependent terms are an (V ) = 0.032(V + 52)/(1 − exp[−(V + 52)/5]) and bn (V ) = 0.5 exp[−(V + 57)/40]. Parameter values are as follows: C = 1 µF/cm2 ; g L = 0.1, g K = 80, g N a = 100 (all in mS/cm2 ); VL = −67, VK = −100, VN a = 50 (all in mV); and I0 = 8 for excitatory cells, I0 = 0.5 for inhibitory cells (both in mA/cm2 ). Each population is heterogeneous in the spontaneous firing rates of the individual cells, with the heterogeneity introduced by varying the value of I0 with a cell-dependent factor, so that neuron i has baseline current αi I0 . This factor is assigned deterministically across the population using a uniform distribution. We parameterize the degree of heterogeneity with a factor H , such that for excitatory

(4)

where A and B are constants giving the rates of rise and decay of the synapse (B = 1/τs , where τs is the synaptic time constant), and S(V ) = 1 + tanh(V /4). When the neuron spikes, V is large and S(V ) saturates at 2, driving s towards s¯ = 2A/(2A + B)  1. Excitatory and inhibitory cells have different rise and decay times: A E = 20, B E = 0.333; and A I = 1, B I = 0.05 (all in ms−1 ). Each neuron influences other cells to which it is connected through a synaptic current of the form Isyn (t) = gsyn s(t − δ)(Vsyn − V ),

and the potassium activation, n, dn = an (V )(1 − n) − bn (V )n. dt

2.1.3. Synaptic Model. Each cell has an associated level of synaptic output, s, governed by

(5)

where s is associated with the presynaptic cell, V is the membrane voltage of the postsynaptic cell, δ represents the axonal delay (if any), and gsyn is the maximum synaptic conductance. For excitatory connections, Vsyn = 0 mV, while for inhibitory synapses Vsyn = −80 mV. All-to-all coupling is implemented by summing the individual synaptic current contributions for every cell in each population. 2.1.4. Noise Inputs. In some simulations, we introduce noise to the system, roughly corresponding to the noise arising from synaptic inputs from neurons outside the model system, arriving at random times and influencing the model neurons. For simplicity, we subject each neuron to an independent noise source, consisting of two independent Poisson processes, one excitatory and one inhibitory. When an excitatory Poisson pulse arrives, the target cell’s voltage is instantaneously incremented by an amount V , and similarly when an inhibitory pulse arrives, the cell’s voltage is decremented by −V . The processes are characterized by the size of the increment (V = 1 mV for both processes), and the frequency of the incoming pulses. When examining the effects of noise on the population (see Section 3), we vary the noise by setting the frequency of Poisson pulses to N ms−1 for both the excitatory and inhibitory processes. Increasing N while keeping V fixed results in a monotonically increasing variance in the compound counting process consisting

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of the sum of the excitatory and inhibitory processes. Numerical simulations (not shown here) confirm that this increased variance translates into a monotonic, though nonlinear, increase in the variance of interspike intervals in a neuron subjected to the two Poisson processes. 2.1.5. Numerical Methods. We use the fourth-order Runge-Kutta method, with a step size of 0.01 ms. Repeating the runs with a smaller step size does not significantly affect the results. 2.2.

Reduction to Map

We seek to extend the work in Ermentrout and Kopell (1998) to include two new features: populations of neurons rather than single cells; and the presence of noise. We will consider the effects of noise below, and for now focus on the implications of replacing single cells with populations. For our heterogeneous populations (heterogeneity in spontaneous firing rates, as described above), not every neuron will participate in every burst of firing (henceforth called a “pulse”; the time associated with each pulse is considered to be the time at which the first neuron fires in the burst). This partial participation fundamentally alters the synchronization behavior of the network, and requires us to expand the previous one-dimensional map to a three-dimensional one: rather than considering only the timing of pulses, we now include both timing and degree of participation. Experimental observations suggest that the first pulse in the doublet is characterized by nearly complete participation of the I-cell population, while the second pulse has only partial participation. We have reproduced this situation in the model, through appropriate selection of synaptic coupling parameters. Numerical simulations show that in this case, the fraction of E-cells participating in each pulse is very nearly constant, but the fraction of I-cells participating in the second pulse of the doublet can vary from cycle to cycle. We therefore write our map in terms of the following three quantities: the difference in firing times between the excitatory populations E2 and E1 (); and the magnitude of the second pulse of the doublet (m 1 and m 2 , respectively, for inhibitory populations I1 and I2). The magnitudes m i are derived from the fractions of cells participating in the second inhibitory pulse of the doublet by using the synaptic time constant to exponentially decrease the contribution to m i of later-firing cells. (Using this correction makes the magnitude re-

flect the net synaptic effect of each burst of firing, which is especially helpful in the presence of noise, when pulses with identical fractions of cells participating may have very different net effects if the time over which the firing is spread out varies.) The map we seek, then, has the form f : [, m 1 , m 2 ] ¯ m¯ 1 , m¯ 2 ], where  = t2 − t1 (with t1 and t2 → [, being the times at which the E1 and E2 populations begin to fire), and m 1 and m 2 are the above-defined magnitudes of the doublets for populations I1 and I2. To formulate the complete map, we consider several submaps, as follows. TI (φ) gives the time to the next pulse in an I population, after receiving two excitatory inputs separated by a time φ. M I (φ) reflects the magnitude of the output of an I population, again after receiving two E pulses a time φ apart. Finally, TE (ψ, m) gives the time to the next firing of an E population, after two inhibitory inputs separated by a time ψ, where the first I pulse has a fixed magnitude (m  1) and the second I pulse has magnitude m. These submaps are evaluated numerically (see Figs. 6A–D and 8A–D) by applying external perturbations with varying timings or magnitudes, while holding all other conditions constant. As shown in Fig. 2, the population firing events are as follows (in roughly chronological order, for the case where E1 leads E2 near synchrony): 1. Pulse e1 occurs at time t1 , with fixed magnitude m E . 2. Pulse e2 occurs at time t2 , also with fixed magnitude mE. e1

e1

E1

i 1L

i 1D

I1

e2

e2

E2

i 2L

i 2D

I2

Time Figure 2. Labels for the firing pulses generated by each population, used in the derivation of the map for pulse timings and magnitudes. Vertical bars represent bursts of firing, arranged in rows associated with each of the four populations; time increases to the right. Populations E1 and I1 constitute a local circuit, as do populations E2 and I2 (see Fig. 1); pulses e1 and e2 elicit pulses i 1L and i 2L , respectively. Distal E-I connections, with axonal delays, elicit pulses i 1D and i 2D . When the inhibition from these pulses wears off, the excitatory pulses are released to fire again, generating pulses e¯ 1 and e¯ 2 .

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3. Population I1 responds to pulse e1 , generating pulse i 1L at time t1 + tei , where tei is the minimum (fixed) time it takes for an excitatory input to elicit an inhibitory pulse. 4. Population I2 responds to pulse e2 , generating pulse i 2L at time t2 + tei . 5. Pulse e1 arrives at I2 at time t1 + δ. The time between the two previous excitatory pulses is ψ2 = t1 + δ − t2 = − + δ, where  ≡ t2 − t1 . Pulse i 2D is generated at time t1 + δ + TI (− + δ), with magnitude m 2 = M I (− + δ). 6. Pulse e2 arrives at I1 at time t2 + δ. Here, we have ψ1 = t2 +δ −t1 = +δ, and pulse i 1D is generated at time t2 + δ + TI ( + δ), with magnitude m 1 = M I ( + δ). 7. Population E 1 is subject to two inhibitory pulses, separated by a time θ1 =  + δ + TI ( + δ) − tei , and pulse e¯ 1 occurs at time t¯1 = t2 + δ + TI ( + δ) + TE (θ1 , p1 ). 8. Population E 2 is subject to two inhibitory pulses from I2 , with a time separation of θ2 = − + δ + TI (− + δ) − tei . Pulse e¯ 2 occurs at time t¯2 = t1 + δ + TI (− + δ) + TE (θ2 , p2 ).

where all partials are evaluated at the fixed point. Evaluating the derivatives, we find:

Using the above timing and magnitude values, we write the complete three-dimensional map as

√ with eigenvalues λ1 = 0, λ2,3 = 12 (a ± a 2 + 8bc). Synchrony is stable when the (possibly complex) magnitude of the eigenvalue pair λ2,3 is less than unity.



   ¯       f : m 1  → m¯ 1  m¯ 2 m2   − + TI (− + δ) − TI ( + δ)  + T (θ (−), m ) − T (θ (), m ) E 2 E 1   =    M I ( + δ) M I (− + δ) where θ () =  + δ + TI ( + δ) − tei . Stability of the Map. We now consider the stability properties of the map, in the vicinity of a synchronous ¯ m¯ 1 , m¯ 2 ) = fixed point, for which (, m 1 , m 2 ) = (, ∗ ∗ (0, m , m ). The Jacobian matrix of partial derivatives is   ¯ ¯ ¯ ∂ /∂ ∂ /∂m ∂ /∂m 1 2   0 0 D f = ∂ m¯ 1 /∂ , ∂ m¯ 2 /∂ 0 0

  ¯ ∂ TE  ∂ TI  ∂ −2 = −1 − 2 ∂ ∂φ φ=δ ∂ψ ψ=θ (0), m=m ∗ 

∂ TI  ≡a (6) × 1+ ∂φ φ=δ  ¯ ∂ ∂ TE  =− ≡b (7) ∂m 1 ∂m ψ=θ (0), m=m ∗  ¯ ∂ ∂ TE  = ≡ −b (8) ∂m 2 ∂m ψ=θ (0), m=m ∗ ∂ m¯ 1 ∂ M I (δ) = ≡c (9) ∂ ∂ ∂ m¯ 2 ∂ M I (δ) =− ≡ −c (10) ∂ ∂ Using the above definitions for a, b, and c, the Jacobian becomes   a b −b   Df =  c 0 0 , (11) −c 0 0

3. 3.1.

Results Stability of Synchrony in Noise-Heterogeneity Space

Figure 3 shows the result of varying the noise input to the system, N , and the heterogeneity of the current inputs to the neurons, H . Briefly, H is a factor multiplying the size of the range of applied input currents, while N gives the frequency of independent Poisson inputs to each neuron; for more detail, see Sections 2.1.2 and 2.1.4. There are several regions of interest in noiseheterogeneity space. On the left side of the plot, the low heterogeneity region (H ≤ 0.6) shows very stable synchrony, not disrupted by even the highest levels of noise applied (though large enough noise inputs will eventually disrupt synchrony even in this region). Point A on the plot marks the zero-noise, zero-heterogeneity case, to be discussed in Section 3.2. In the center of the plot (0.8 ≤ H ≤ 1.8) is a region exhibiting the

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ity always reduces the level at which noise-stabilized synchrony is lost by disruption from high noise. The latter, however, is not the case: at some points in the space, increased heterogeneity makes the system easier to stabilize with low noise (H in the range 0.8 to 1.2), while in others increased heterogeneity makes it more difficult to stabilize the system (H in the range 1.4 to 1.6). In this work, we focus on representative points in each region, and apply our map approach at each point.

−1

Noise, N [Poisson frequency, ms ]

2

D

1.5

1

C

0.5

3.2. 0

Stable Synchrony (N = 0, H = 0)

B

A

0

0.2

0.4

0.6 0.8 1 1.2 1.4 Heterogeneity, H [scaled]

1.6

1.8

2

Figure 3. A plot showing the stability of synchrony in noiseheterogeneity space. Stable synchrony (or near-synchrony) is marked with an ‘◦,’ and unstable synchrony is marked with an ‘×.’ Heterogeneity, H , is a factor affecting the size of the range of applied currents in the excitatory and inhibitory populations (see Section 2.1.2). Noise, N is varied by increasing the frequency of the excitatory and inhibitory Poisson inputs independently applied to each neuron (see Section 2.1.4). Note the presence of a region (0.8 ≤ H ≤ 1.8) displaying noise-stabilized synchrony exists: for low noise levels, synchrony is unstable, but a moderate noise level acts to stabilize the synchronous state. The points enclosed by squares and marked with letters are discussed in the text, as follows: point A, Section 3.2; point B, Section 3.3; point C, Section 3.4; and point D, Section 3.5.

phenomenon of primary interest to us, noise-stabilized synchrony. For intermediate levels of heterogeneity, synchrony is unstable at low levels of noise, then becomes stable for a range of moderate noise levels, before finally finally losing stability for large noise inputs. This central region thus has three parts, and we will examine a representative point in each: heterogeneity disrupting synchrony for low noise (point B on the plot, see Section 3.3); noise-stabilized synchrony (point C, Section 3.4); and large noise disrupting synchrony (point D, Section 3.5). Finally, at the far right side of the plot, heterogeneity is so large that noisestabilized synchrony does not occur; the system passes directly from a region in which heterogeneity disrupts synchrony to one in which noise disrupts synchrony, with no stable regime between the two. The shape of the region of noise-stabilized synchrony reveals a somewhat complicated relationship between noise and heterogeneity. Intuitively, we might expect that increasing heterogeneity should make synchrony easier to disrupt with high noise inputs, and more difficult to stabilize with low noise inputs. The former appears to be correct: increasing heterogene-

In the absence of heterogeneity, each population is equivalent to a single cell, and thus this point in noiseheterogeneity space is the same as one of the cases examined in Ermentrout and Kopell (1998), with δ = 10 ms. As in that earlier work, synchrony is stable in this case, as may be seen in Fig. 4; the system is started in perfect synchrony, and after an external perturbation is applied at t = 60 ms, synchrony is quickly restored. The external perturbation consists of a mild inhibitory pulse (synaptic conductance gext = 0.5 mS/cm2 ) applied to E1. Note that in this case, as in all following cases, the nature of the perturbation is not critical, and the same stability (or instability) of synchronization is seen whether excitatory or inhibitory perturbations are applied to either the excitatory or inhibitory populations. In this case, the three-dimensional map formulated in Section 2.2 is reduced to a single dimension, since the quantities m 1 and m 2 do not vary from cycle to cycle; that is, b = c = 0 in Eqs. (7)–(10). The single eigenvalue is then λ = a ≡ −1 − 2



∂ TI ∂ TE ∂ TI −2 × 1+ , ∂φ ∂ψ ∂φ

(12)

where all derivatives are evaluated at the fixed point. Numerically evaluating the maps for this case shows that ∂ TI /∂φ  0 and ∂ TE /∂ψ  −0.3, indicating that λ  −0.6. Since |λ| = 0.6 < 1, the map correctly predicts stable synchrony for this case. Moreover, comparing successive values of  ≡ t2 − t1 in runs such ¯ as that shown in Fig. 4 indicates that / fluctuates approximately in the range −0.6 to −0.5, a reasonable match to the value of the eigenvalue. As an alternate means of achieving stable synchrony in the presence of heterogeneity, we consider a case in which the distal, axonally delayed coupling is strong enough to cause all cells in the inhibitory population to

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Figure 4. Stable synchrony, with no heterogeneity and no noise. This raster plot marks the firing time of each neuron in each population. Top: Excitatory populations, E1 and E2. The system is considered synchronous when these populations fire simultaneously. Bottom: Inhibitory populations, I1 and I2. The system is started in synchrony, then perturbed at t = 70 ms, after which it quickly recovers synchrony. Parameters given in Section 2.

participate in the second pulse of the doublet. Setting the distal coupling conductance equal to the local coupling, cei = gei = 0.2 mS/cm2 , numerical simulations (not shown) demonstrate that synchrony is stable under these conditions. With this strong distal coupling, the three-dimensional map is again reduced to a single dimension, since the quantities m 1 and m 2 do not vary from cycle to cycle when the system is near synchrony. Numerically evaluating the maps for this case shows that ∂ TI /∂φ  −0.04 and ∂ TE /∂ψ  −0.27, indicating that λ  −0.4. Since |λ| = 0.4 < 1, the map correctly predicts stable synchrony for this case. Comparing successive values of  ≡ t2 − t1 in numer¯ ical simulations indicates that / fluctuates approximately in the range −.37 to −.43, a good match to the predicted value of the eigenvalue. We note that all cases with distal E-I coupling strong enough to induce consistently complete participation by the I-cells in the second pulse of the doublet are seen, numerically, to be stable. This is consistent with our finding that it is the effect of partial participation of the I-cells which is responsible for destabilizing synchrony in the noise-free case; see the Discussion for more detail.

3.3.

Unstable Synchrony (N = 0, H = 1)

In the cases discussed in Section 3.2, all inhibitory cells participate in both pulses of the doublet. This situation is not thought to occur experimentally (M. Whittington, pers. comm; Whittington et al., 2000); rather, there is incomplete participation by the I-cell population on the second spike of the doublet. To reflect this, we consider a heterogeneous case (H = 1, as defined in Section 2.1.2) where the distal excitatory coupling does not elicit a full response from the I-cells, by setting cei = gei /2 = 0.1 mS/cm2 . In this case, synchrony is no longer stable, as seen in Fig. 5, where an external perturbation introduced at t = 65 ms drives the system quickly away from its initially synchronous state. As above, the external perturbation consists of a mild inhibitory pulse applied to E1 (synaptic conductance gext = 0.25 mS/cm2 ; a smaller perturbation was used in this case to reduce the speed at which the instability grew). Analysis of the map indicates that there is one eigenvalue with a value of zero (see Section 2.2), and thus a perturbation in the direction of the stable eigenvector could potentially maintain synchrony in this case. Such a perturbation, however, would require

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Figure 5. Unstable synchrony, with heterogeneity H = 1 (see Section 2.1.2) and zero noise. This raster plot marks the firing time of each neuron in each population. Top: Excitatory populations, E1 and E2. The system is considered synchronous when these populations fire simultaneously. Bottom: Inhibitory populations, I1 and I2. The system is started in synchrony, then perturbed at t = 65 ms, after which synchrony is quickly lost. Parameters given in Section 2.

precisely-timed inputs to the excitatory and inhibitory cells, chosen such that they altered the map variables , m 1 , and m 2 exactly along the eigenvector direction; the occurrence of such a constrained set of stimuli is biologically implausible, and thus we consider only the case of one-time perturbations to either the E-cell or the I-cell populations. To evaluate the stability of the map in this case, we first numerically obtain the individual submaps and their partial derivatives at the fixed point. Figure 6A shows TI (φ); Fig. 6B shows M I (φ); Fig. 6C shows TE (ψ, m) with m fixed and ψ varying; and Fig. 6D shows TE (ψ, m) with ψ fixed and m varying. Using these plots, we find the following partial derivatives, evaluated at the fixed point:

∂ TI  −0.2, ∂φ ∂ MI  0.08, ∂φ ∂ TE  −0.66, ∂ψ

(13)

and ∂ TE  21. ∂m

(16)

Substituting these derivatives into the previous definitions for the quantities a, b, and c (see Section 2.2), we find a  0.46, b  −21, and c  0.08. Using these quantities to evaluate the eigenvalues of the Jacobian matrix, we find

1 a ± a 2 + 8bc 2  0.23 ± 1.8i.

λ2,3 =

(17)

The magnitude of this complex conjugate pair of eigenvalues is |λ2,3 |  1.8 > 1, showing that the map correctly predicts unstable synchrony for this case. 3.4.

Noise-Stabilized Near-Synchrony (N = 0.5, H = 1)

(14) (15)

The lack of stability in the above case is surprising, since experimentally it appears that the system can maintain synchrony despite partial participation of

Noise-Stabilized Long-Distance Synchronization in Model Neurons

TI(φ) [ms]

A

8

I

7

M (φ) [ms]

0.6 9

B

0.4 0.2

6 8

10 12 φ [ms]

0

14

6

8

10 12 φ [ms]

14

C

18

E

16

T (ψ,m) [ms]

19

20

E

T (ψ,m) [ms]

6

10

12 14 ψ [ms]

16

18

To understand the source of this stabilization in the presence of noise, we turn once again to the timing and magnitude maps, numerically evaluating them in the noisy case. The maps are noticeably altered by the presence of noise (see Fig. 8), leading to the following revised partial derivatives at the fixed point (note that we report several digits in these numbers, but after the intermediate calculations we will round to one significant figure):

D

17 16 15

151

0.3

0.35 0.4 m

0.45

Figure 6. Submaps used to evaluate the complete map, deterministic case. Model parameters are as described in Section 2. A: The submap TI (φ), giving the time to the next pulse in an I population, after receiving two excitatory inputs separated by a time φ. B: The submap M I (φ), giving the magnitude of an I population’s output, after receiving two E pulses a time φ apart. C: The submap TE (ψ, m), giving the time to the next firing of an E population, after two inhibitory inputs separated by a time ψ, where the first I pulse has a fixed magnitude and the second I pulse has magnitude m. Here, we vary ψ while keeping m fixed at m = m ∗ , where m ∗ = M I (δ) = 0.38 is the doublet magnitude observed at perfect synchrony. D: The submap TE (ψ, m), now varying m while ψ is held fixed at ψ = ψ ∗ , where ψ ∗ = 13.7 ms is the interval between inhibitory inputs observed when the system is in perfect synchrony.

the I-cells on the second pulse of the doublet. One significant omission in the simulations described in Sections 3.2 and 3.3 is the effect of neurons not explicitly modeled in the simulation. Pulses from cells outside the scope of the simulation can be expected to arrive at effectively random times, influencing the cells and acting as a source of noise in the system. Significantly, when we add such noise (in the form of equal-rate excitatory and inhibitory Poisson processes, as described in Section 2.1.4), we recover stable synchrony, as shown in Fig. 7. Note that the influence of the noise means that synchrony is never perfect in this case: the system actively maintains a nearly synchronous state, but with cycle-to-cycle variations induced by the noise. (Tiesinga and Jos´e (2000) refer to a related phenomenon as “stochastic weak synchronization.”) Figure 7 shows that when an external perturbation simulating a mild inhibitory pulse (gext = 0.7 mS/cm2 ) is applied to E1 at t = 60 ms, the system quickly returns to near-synchrony. Excitatory and inhibitory inputs to either E-cell or I-cell populations all show a rapid recovery of near-synchrony.

∂ TI  −0.07 ± 0.46, ∂φ ∂ MI  0.027 ± 0.017, ∂φ ∂ TE  −0.90 ± 0.10, ∂ψ

(19)

∂ TE  18.7 ± 2.6. ∂m

(21)

(18)

(20)

and

In the presence of noise, the timing and magnitude values vary significantly from run to run, and the above errors represent one standard deviation in the data, where a minimum of 100 samples have been taken at each data point. Substituting the above values into the definitions for a, b, and c, and propagating the errors using standard methods (Taylor, 1997), we find a = 0.81 ± 1.25, b = −18.7 ± 2.6, and c = 0.027 ± 0.017, from which the complex conjugate pair of eigenvalues is

1 a ± a 2 + 8bc 2  [0.40 ± 0.61] ± [0.92 ± 0.45]i.

λ2,3 =

(22)

The magnitude of this pair of eigenvalues is then |λ2,3 |  1 ± 0.5. The map thus predicts an average value on the very edge of stability, but the noise in the system creates a large uncertainty in the size of the eigenvalues. Significantly, the size of the eigenvalues has been reduced substantially, from 1.8 in the deterministic case to 1 in the presence of noise. The dominant factor in the size of the eigenvalues is the product bc; the deterministic and noisy cases have comparable values for b ≡ −∂ TE /∂m, while c ≡ ∂ M I /∂φ is significantly reduced in the noisy case.

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Figure 7. Noise-stabilized synchrony, with heterogeneity H = 1 (see Section 2.1.2) and noise N = 0.5 (see Section 2.1.4). This raster plot marks the firing time of each neuron in each population. Top: Excitatory populations, E1 and E2. The system is considered synchronous when these populations fire simultaneously. Bottom: Inhibitory populations, I1 and I2. Because of the influence of the noise, synchrony is never perfect, but when the system is started near synchrony and perturbed at t = 60 ms, we see that near-synchrony is quickly restored. Parameters given in Section 2.

3.5.

Synchrony Destabilized by High Noise (N = 1.6, H = 1.8)

The phenomenon of noise-stabilized synchrony does not persist to arbitrarily large noise levels (see Fig. 3). Rather, at some point (varying depending on the degree of heterogeneity present in the system), the noise becomes sufficiently large that the stability is disrupted. Figure 9 shows an example of a numerical simulation with high heterogeneity (H = 1.8) and high noise (N = 1.6); the system is initialized in perfect synchrony, and despite the absence of external perturbations, the noise quickly disrupts the synchronization. Similar behavior is seen at all points in the upper righthand region of Fig. 3. The map approach employed in previous sections breaks down in this region of noise-heterogeneity space. Evaluating the map eigenvalues relies on the ability to evaluate partial derivatives for the various submaps in the vicinity of a synchronous solution. In a purely deterministic case, this solution may unstable (as in Section 3.3), since it is possible to obtain deriva-

tives using the artificial situation in which all neurons are initially in perfect synchrony; with no noise or external perturbations, the system maintains synchrony even though the solution unstable. In the current case, the noise quickly destroys synchrony even with initially perfect synchronization. Thus, the synchronous solution about which partial derivatives are to be obtained cannot be sensibly defined, and therefore the map approach is no longer applicable. 4.

Discussion

We present the following heuristic version of our map analysis to clarify how the heterogeneity-induced destabilization of synchrony occurs. Consider the nearsynchrony case in which E1 leads E2: using the labels in Fig. 2, pulse e1 occurs before pulse e2 . Each of the populations I1 and I2 receive two excitatory inputs, but the time between these inputs differs: I2 receives its second (axonally delayed) excitatory pulse sooner after the first pulse of its doublet than does I1. As discussed in Ermentrout and Kopell (1998), with

8

A

I

7

M (φ) [ms]

TI(φ) [ms]

Noise-Stabilized Long-Distance Synchronization in Model Neurons

6 8

B

0.2 0

12

8

10 φ [ms]

12

18

E

C

T (ψ,m) [ms]

TE(ψ,m) [ms]

18 16 14 12 10 8

10 φ [ms]

0.4

13 14 15 16 17 ψ [ms]

16

D

14 12 10

0.2

m

0.4

Figure 8. Submaps used to evaluate the complete map, noisy case (N = 1, see Section 2.1.4). (Dashed lines) The deterministic submaps, shown for comparison. (Solid lines) Noisy submaps, with error bars indicating one standard deviation over at least 100 independent trials. A: The submap TI (φ). B: The submap M I (φ). C: The submap TE (ψ, m), varying ψ while keeping m fixed at m = m ∗ ; note that m ∗ = M I (δ) = 0.14 differs from the value used in the deterministic case (Fig. 6C, m ∗ = 0.38), since the doublet magnitude is systematically lower in the presence of noise. D: The submap TE (ψ, m), now varying m while ψ is held fixed at ψ = ψ ∗ ; note that ψ ∗ = 13.7 ms differs slightly from the value used in the deterministic submap (Fig. 6D, ψ ∗ = 13.4 ms) because of slight variations in the pulse timings in the two situations. (C and D) Note that because of the effects of noise, both ψ and m vary from cycle to cycle. In generating these plots, m ∗ and ψ ∗ have been set to typical values.

single cells (or equivalently populations with zero heterogeneity and zero noise) this leads to a situation in which the E cells exchange leads on successive cycles, so that in the present example, E2 would lead E1 on the next cycle; for stable synchronous solutions, this process of exchanging leads approaches synchrony geometrically. Now consider what happens when we examine heterogeneous populations, in which the magnitude of the second pulse in the doublet is free to vary. Population I2 receives its second excitatory input sooner after its last burst of inhibitory firing than population I1, and thus there is more self-inhibition present (from the local I-I connections) when I2 receives its excitatory input than when I1 receives its excitation. The cells in I2 are less prepared to spike than those in I1, because of this increased self-inhibition, and thus the magnitude of pulse i 22 is less than the magnitude of pulse i 12 ; this is reflected in the map formulation by the fact that ∂ M I /∂φ > 0, as seen in Fig. 6B. The reduced magni-

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tude of I2’s inhibitory pulse then delays the next firing of E2 less than I1’s inhibition delays the firing of E1 (since ∂ TE /∂m > 0, see Fig. 6D). Recall that in the single-cell system, E2 would have led E1 on the next cycle in this situation; now, with the varying magnitudes of the inhibitory pulses, E2 leads E1 more than in the single-cell (zero heterogeneity) case, and thus synchrony is less stable with the heterogeneous population than it would have been in the single-cell (zero heterogeneity) case. When the effect of varying magnitudes is sufficiently large, it pushes the population into a region in which synchrony is unstable. Note that the above argument indicates that the ability to vary the fractions of cells participating in the second pulse of the doublet will always decrease the stability of synchrony in networks with the architecture we consider here, provided that reducing the time separation of successive excitations leads to a reduced fraction of cells participating; that is, provided that ∂ M I /∂φ > 0. We would expect the latter to be the case in most physically realistic dynamical models, since cells generically become less refractory as the interstimulus time increases, whether that refractoriness arises from self-inhibition (as in our current case), or from some other cause. Thus, we do not expect that consideration of alternative model equations would change our central results. The above heuristic argument also sheds some light on how the presence of noise acts to restore the stability of synchrony. Figure 8B shows that, when noise is added, the derivative ∂ M I /∂φ is significantly reduced, and thus the effect of varying magnitudes is less substantial than it would be without the noise. In a deterministic heterogeneous population, the fraction of inhibitory cells firing is entirely a function of the timing of the excitatory stimuli, and the cells fire in a stereotyped manner, with the inherently fastest cells firing first, and all others following, until the accumulated self-inhibition prevents the slower cells from achieving their firing thresholds. With noise, the voltages of the cells at the instant when an excitatory stimulus arrives are not stereotyped, and even cells with slow inherent rates may, by chance, be close to threshold and thus able to fire. The fraction of cells which fire is therefore a much weaker function of the stimulus timing, which manifests itself in the reduced slope ∂ M I /∂φ and, in turn, in the improved stability of synchrony in the presence of noise. The stabilizing effect of noise in our model suggests that the corresponding biological system may benefit

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Figure 9. High noise destabilizing synchrony, with heterogeneity H = 1.8 (see Section 2.1.2) and noise N = 1.6 (see Section 2.1.4). This raster plot marks the firing time of each neuron in each population. Top: Excitatory populations, E1 and E2. Bottom: Inhibitory populations, I1 and I2. After being initialized in perfect synchrony, the noise quickly drives the system to an asynchronous state. Parameters given in Section 2.

from its exposure to noise. Other beneficial effects of noise have been studied in recent years, in a number of nonlinear and biological systems. Considerable research effort has been devoted to the phenomenon of stochastic resonance (SR), reviewed in Wiesenfeld and Moss (1995), Gammaitoni et al. (1998) and Collins (1999), in which the transmission of a subthreshold stimulus is enhanced by the presence of an optimal level of noise; SR has been experimentally demonstrated in a number of neural systems (Douglass et al., 1993; Collins et al., 1996a, b; Richardson et al., 1998; Russell et al., 1999; Stacey and Durand, 2001; Manjarrez et al., 2002). Biochemical processes taking place inside living cells are subject to substantial fluctuations, and there has recently been speculation that cells may exploit this noise to enable them to choose between competing developmental pathways (Arkin et al., 1998), that high noise levels may reduce the energetic cost of protein production (Thattai and van Oudenaarden, 2001; Ozbudak et al., 2002; Hasty and Collins, 2002), and that spontaneous fluctuations may act to reduce the level of uncertainty in regulated processes through a mechanism dubbed “stochastic focusing” (Paulsson et al., 2000).

Noise has been shown to enhance the synchronization of coupled chaotic systems (Maritan and Banavar, 1994; Lai and Zhou, 1998; Kocarev and Tasev, 2002; Zhou et al., 2002; Zhou and Kurths, 2002). Such studies have generally focused on the case where all of the individual chaotic systems receive the same noise input, though Zhou et al. (2002) observed enhanced synchronization behavior even when using independent noise sources; recall that we also apply independent noise inputs to each neuron. Zhou et al. (2002) consider a case in which the coupling is not strong enough to fully lock all of the unstable periodic orbits (UPOs) of their chaotic system, and note that “phase slips” involve the system following an unlocked UPO for some time. They attribute the positive effect of noise to its ability to reduce how often the system follows the unlocked UPOs long enough for phase slips to occur. This is a very different effect than the one described in the present paper, where the noise operates by delaying a bifurcation to instability. A model neural system in which noise acts to delay a bifurcation has been studied by Laing and Longtin (2001). They examined the stability of localized regions of actively firing neurons (“bumps”), noting that

Noise-Stabilized Long-Distance Synchronization in Model Neurons

while augmenting the neural dynamics with spikefrequency adaptation could destabilize the bumps (that is, cause a previously stationary solution to move), the addition of noise could reduce the average speed of the moving solution to nearly zero, effectively restabilizing the bump. The onset of motion arises through a supercritical pitchfork bifurcation in the bump speed, and the introduction of noise effectively delays the onset of this bifurcation, increasing the range of parameters for which the bump is nearly stationary. Their mechanism of noise-enhanced bump stability is thus conceptually similar to that underlying our noise-induced synchrony. Other recent studies of model neural systems have specifically examined noise-induced synchronization. Wang et al. (2000) considered a population of HodgkinHuxley neurons in the excitable regime (that is, not spontaneously firing), coupled through global excitation; they found that for sufficiently strong coupling, the presence of an optimal level of noise induced near-synchronous firing in the population. Hu and Zhou (2000) observed similar noise-induced nearsynchronization when considering a ring of FitzHughNagumo neurons in the excitatory regime: for sufficiently strong coupling strengths, the addition of noise led to synchronous firing in the ring of model neurons. In both of these systems, oscillations are induced by the noise itself, and the coupling acts to synchronize the motion of the neurons around the noise-induced limit cycle. This differs from our system, in which all neurons fire spontaneously, with the noise simply modulating their firing times. Tiesinga and Jos´e (2000a) obtained noise-induced stochastic synchronization in a network of Hodgkin-Huxley neurons augmented with calcium currents so that they displayed a post-inhibitory rebound effect. They found a threshold synaptic strength below which no stable spiking states existed, and observed that noise could sustain a stochastically synchronized spiking state even below this synaptic strength threshold. Note that all of the above studies consider single populations of neurons, as distinct from our set of multiple interacting populations. Studying single cells, Sosnovtseva et al. (2002) observed noise-induced stochastic synchronization in a set of two E-cells each connected to the same I-cell; each cell had Hodgkin-Huxley dynamics augmented with a mechanism for afterhyperpolarization. Other work has addressed the effects of both heterogeneity and noise on synchrony in neural systems.

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Tiesinga and Jos´e (2000) examined a single population of inhibitory neurons, incorporating both heterogeneity (introduced in the current applied to the neurons, as in our case) and noise (in the form of independent Gaussian currents applied to each neuron). They found that in cases of weak current heterogeneity, the degree of synchronization was improved by the addition of noise; as in our case, the synchronization thus obtained was approximate, with not every cell firing on every cycle (as previously noted, they refer to this situation as “stochastic weak synchronization”). Karbowski and Kopell (2000) studied a network of local circuits, each of which had the same E-I connectivity shown in our Fig. 1. They noted that introducing a small amount of disorder in the axonal time delays between circuits improved the stability of the synchronous solution in which each inhibitory population fired doublets; note that this is quite a different type of heterogeneity from the varying current inputs considered here. With this form of heterogeneity, no beneficial effects of noise were observed: noise acted only to reduce the stability of the doublet-based synchronous solution. The numerical simulations we have employed in this work are 800-dimensional (400 cells with two dynamical equations each), whereas the map analysis is carried out in three dimensions. This is a substantial simplification, and it is encouraging that the map approach is able to predict the stability properties of the full system. Having established the utility of the approach, it would be interesting to relax some of our simplifications and consider how such changes affect the predicted and observed stability properties. For example, additional currents implementing effects such as after-hyperpolarization or spike-frequency adaptation would have implications for the refractory characteristics of the cells. Since our heterogeneous populations lost stable synchrony (in the absence of noise) largely through a refractoriness effect, we might expect such extra currents to have interesting stability implications, both with and without the addition of noise. The noise itself has been applied in a relatively simple form, and a more detailed study might focus on the relationship between the characteristics of the noisy inputs and the effects on stability. Finally, the relationship between noise and heterogeneity remains to be explored in more detail, so that the shape of the region of noise-stabilized synchronization (shown in Fig. 3) may be understood in more greater detail.

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