characterized, we move on to mapping the critical value of τI at which the transition occurs.
Motivated by the understanding gained from the approach of thechapter 2, we apply the same ideas here, providing a prediction of the critical value as a function of K and ¯ν. Then, in an attempt to understand the fate of flux tubes across the transition we are lead to explain how the distance time series between trajectories initially separated by some finite value are qualitatively altered as τI increases. From these observations, we can identify a microscopic destabilizing mechanism likely responsible for the instability which dominates the stability behaviour once the stabilizing effect of the pulsed inhibition is removed beyond the critical τI. This chapter concludes with an outlook on the transition out of stable chaos.
4.3 The correlated Leaky Integrate-and-Fire (cLIF)
The equations for the leaky integrate-and-fire neuron model with exponentially decaying synap-tic currents (cLIF), are obtained from the 2D linear modelEquation 3.4when the voltage does not feedback to the current, g = 0, and the input is fed only into the current, γ = 0, such that the auxiliary current becomes a low pass filtering synapse. Replacing the symbol for the auxiliary current W with I for synaptic current, the dynamical equations of the system are
τvv˙i =−vi+I+Irheo+Iext
v. From our general propagator solutionEquation 3.17 for real eigenvalues S(∆t) =er∆t{cosh(ω∆t)I+ sinh(ω∆t)(A−rI)/ω} So that the equations reduce to
v(t+ ∆t) = (v(t)−Iext)e−
As with the LIF, the spike is approximated by a hard voltage threshold at vT = 1, whose crossing by the state leads to a discontinuous reset of the voltage to a reset value vR= 0.
The synaptic current can be normalized to keep either the mean or variance of the input fixed, but not both. By placement of τv as a prefactor in the input to I, we are normalizing by the mean input current.
J ˆ ∞
0
τv τI
e−τIt dt=J τv ,
which when integrated over v normalizes to J, the total effect on the voltage as in the case of the LIF considered in chapter 2. Together, the many, uncorrelated inputs to I resemble a white noise Gaussian process, a limit that is precisely the diffusion approximation discussed in the next chapter. Such a process has a constant power spectrum. Thus, I is approximately an Ohrenstein-Ulhenbeck process, whose statistics are easily handled analytically. Its power spectrum is obtained for its Fourier domain solution, from which we can use the Wiener-Khinchine theorem to get its autocorrelation function,
C(τ) : = J0τv
√KτI
!2
Kν¯ ˆ ∞
−∞
dt 1
1 +ω2τI2eiωt C(τ) = J02 τv2
2τIν¯exp(−|τ| τI ) so that the current variance is
σI2 =J02 τv
2τIντ¯ v . (4.3)
To normalize instead by this variance, we would replace the τv prefactor in Equation 4.1 with
q2τv/τI, makingσI2 =J02ντ¯ v, matching the variance of the LIF. We will return to this seemingly innocuous choice of normalization when we consider the large-K behaviour of the stability in subsection 4.7.2.
We note that there are other temporally extended kernel functions, aside from the exponential one used here, that could be used for the synaptic current. A notable example is theα-function studied in a pair of papers[14,73], that is obtained from a normalized difference of exponentials, with characteristic times τI,1 and τI,2 by taking the limit τI,1 →τI,2. It allows additionally for a finite rate of rise into the voltage, which then integrates to a smooth voltage (the step rise in the current considered here integrates to a voltage kink). The effects that we will study arise from sharp rises in current, which both models allow, so we expect our results to generalize to that model. However, one difference between that study and here is that there they have have fixed the number of connections entering any neuron, while in our case we only fix the mean.
A discrepancy in the observed N-scaling of λmax between the two models may arise from this difference. In particular, the cLIF value was observed [12,11] to scale as logN while the value obtained from the α-function model was found to converge exponentially to a finite value. We conjecture that the discrepancy is due to the silent fraction existing in our networks and not in theirs, whose size grows with logN. Nevertheless, given that thisα-function kernel brings with it the added complications of higher order differential equations and more complex conditions, we keep with the simple low pass integrator model for the synaptic current.
Also note that the cLIF solution,Equation 4.2, is not generally invertible. Monteforte calculated a few of the special cases in which it is invertible. These exist by setting the value of synaptic time constant to a low-valued integer multiple or fraction of the membrane time constant.
4.3 The correlated Leaky Integrate-and-Fire (cLIF)
The voltage solution is then a polynomial in the exponential of the time and can be solved analytically using algebraic techniques. Such special cases do not suffice for comprehensively tracking the transition to chaos asτI varies, however. As that is the topic of this chapter, we use the 2D linear model exposed inchapter 3for whichτI varies smoothly and over an unrestricted range. Since the implementation of that model and of Monteforte’s special cases are completely different, however, a comparison the results of the two models serves as a consistency check when the 2D linear model is set to these special cases. We give a comparison of spectra obtained both ways in Figure 4.1, both to show consistency and to show the qualitative nature of cLIF spectra. The correspondence is exact to the desired precision, limited only by the machine.
0 100 200 300 400
0 500 1000 1500 2000 2500 3000
10−20
Figure 4.1: Consistency check of implementation by comparison with another. (a) cLIF spectra of our implementation (black) shows consistency with another implementation(red dashed). Black dashed line is−1/τI, dotted line is −1/τv. (b) Convergence with the simula-tion window of the 0-exponent,λ0 for the two stable cases shown in (c). (c) The amplification (or not) of finite machine precision deviations due to chaotic (or stable) dynamics, respec-tively. (N = 200, K = 100)
Note that the spectra have two lobes, each with a plateau, one at τv and the other at τI and that the last exponent is separated from the rest. We also show that machine precision differences in the numerics are amplified in the chaotic regime of the dynamics such that the simulations of the formally identical implementations diverge from each other after some time.
Simulations of chaotic dynamics will always diverge from the physical trajectory started at the same initial condition due to finite precision. Nevertheless, for a wide class of systems, the numerical trajectory is shadowed by some other physical trajectory and thus remains relevant to the system [74]. We will assume that the networks considered here exhibit this property. In the distance time series, we can also see what appears to be the finite-size spiking instability discussed in the previous chapter take hold at around the time when the spike time difference reaches 10−5. We will discuss the effects of this instability later in the chapter. We only mention here that it has a non-negligible contribution to the chaotic dynamics. The two trajectories
nevertheless stay within machine precision of each other when simulated in the stable regime, as expected. The demonstrated accuracy of both implementations can be seen in the convergence of the zero-exponent, λ0, which converges to 0 with simulation time as required.
Since Monteforte gave some results for this model in his PhD thesis, we end this preliminary section by listing them. In the range ofτIthat he accessed (τI/τv = 1/3,1/2,2,3), he found that the maximum Lyapunov exponent, λmax ∝ logN. He also found that the entropy production and the attractor dimension, H, D ∝ N (with a rather small coefficient) so that the chaos is extensive. Definingh:=H/N and d:=D/N, he found thatλmax, h,d converge with K. λmax and h also converged with ¯ν. A mean field approximation of the mean Lyapunov exponent can be made for the cLIF similar to that exposed in chapter 2for the LIF. For currents much larger than Irheo, the corresponding integral over the states vanishes and the result simplifies toλmean∼ −12τ1
v + τ1
I
, whose approximation appears to be much better at moderately sized K than for the finite-K approximation of the mean exponent of the LIF. It holds valid even in this chaotic regime. We will show that some of these scaling properties change at large τI. Motivated to understand the novel feature of correlated activity in the cLIF compared to the LIF, arising from the ongoing dynamics of the synaptic currents, we begin the results of this chapter in the next section by situating the dynamics of balanced networks of cLIF neurons along the axis of the strength of recurrent interactions.