Distance time series fluctuations near the transition

With the synaptic current as an additional degree of freedom in the single neuron dynamics there is some choice now compared to the LIF case as to how both the perturbation is applied and the resulting distance measured. Since there is no simple phase representation, we must perturb in the voltage and current representation. In that space, the trajectory vector is state-dependent and so must be calculated specifically for the state being perturbed in order to ensure the perturbation vectors lie in the plane orthogonal to the trajectory so that all of the perturbation strength is mapped into relative changes in the spike time sequence, and not lost to global shifts. Indeed, the trajectory vectors for perturbed states will now be angled away from that of the state being perturbed, in contrast to the case of the phase representation of the LIF in which they are all parallel. We have checked this angle Figure 4.14. In particular, one sees that the continuous flow in the full space has a local divergence whose rotational symmetry is increasingly broken with τ_I. This can be explained, at least partially, through the effect of τ_I on the eigenvectors of the single neuron dynamics, which we will investigate in more detail in a later section. The angles are nevertheless small.

~z_⊥1

~z⊥2

1−~z_ref·~z_perp

−0.2 −0.1 0 0.1 0.2

−0.2

−0.1 0 0.1 0.2

Figure 4.14: Anisotropy in the local divergence of the flow increases with τ_I. Shown is a contour plot of the degree of separation of tangent vectors, ~z_perp, of trajectories local to the unperturbed trajectory in a plane orthogonal to the tangent vector, ~z_ref, of that trajectory and spanned by ~z⊥1 and ~z⊥2. The separation is quantified by the deviation from 1 in the mutual overlap between ~z_perp and ~z_ref. The contour values increase from 0 away from (0,0) and range up to∼10⁻⁵ at the edges of the plot, with larger values for largerτ_I. The contour color (ranging from blue to red) denotes the set of contours for a given τ_I fromτ_I = 1ms to τ_I = 10ms.

The difference between the voltage and current subspace with regards to where the perturbation is applied comes down to the additional filtering of the perturbation through the current. This adds an additional layer of complication and so here for simplicity we perturb only in the voltage subspace.

The quantitative effects of the perturbation in the distance depend on which space that distance is calculated in. The only qualitative difference between the two choices is that over periods when the index sequences are aligned the distance evolution between spike times can grow or

decay in the voltage subspace, while it always decays in the current subspace. We only consider the voltage subspace. The decorrelation criteria is relatively independent of the choice of the space since the voltage and the current both undergo a decorrelation. We continue to use the 1-norm, normalized by N, to measure the distances so that the magnitudes are relative to the distance between the resting and threshold voltage, v_T −v_R= 1.

Figure 4.15: (A) distance times series exhibits transient amplification whose strength and length increases with τ_I from blue to red approaching the transition. (τ_I values from Fig-ure 4.13)

InFigure 4.15, we show a representative set of distance examples for values ofτ_I asymptotically approaching the critical value from the stable side. They were obtained from perturbation strengths just below the one at which they diverged. We make a few qualitative observations from these examples. Their clarification sets the agenda for the remainder of this chapter.

First, the decay of the distance far from the transition is well approximated by the mean Lyapunov exponent. Second, the ‘spikes’ observed in distance are periods over which the spike sequence of the perturbed trajectory is advanced or delayed by one spike relative to the unperturbed trajectory. Once the spike sequences align again, the distance falls back down.

However, the evolution of the distance on these spikes is now qualitatively different than for the LIF. Third, as the transition is approached withτ_I and the long term decay slows; on the short term, there appear stronger, temporally correlated fluctuations in the decay of the distance.

From this third observation, we highlight an important complication arising from long obser-vations of the strongly fluctuating distance. Namely, the distance can have decayed to machine precision, where it remains, until experiencing a sufficiently strong and long fluctuation over which it can grow past the characteristic distance where the finite-size spiking instability takes hold, leading it to then diverge. This fluctuation, however, may not have led to such a diver-gence had the precision of the machine been higher and the distance allowed to decay further before experiencing the fluctuation. Thus, reliable results can only really be made over the subset of trajectories that have not yet dropped to machine precision.

4.8 Microstate analysis of the transition to chaotic dynamics

10⁻¹⁰ 10⁻⁵ 10⁰

0 0.2 0.4 0.6 0.8 P(e)

10⁰ 10² 10⁴ 10⁶

−5 0 5

s

2

λ1,λ2

(a) (b)

(c)

e

Figure 4.16: (a) P_s() for τ_s approaching the critical transition (dashed lines are at double precision; lines are at long double precision). (b) _crit and t_decorr are anti-correlated. (c) Convergence of λ_max(solid lines) and λ₀(dotted lines) for τ_I = 7ms < τ_I^crit(blue) and τ_I = 8ms> τ_I^crit(red) .

However, this subensemble of trajectories produces a bias in any averaging towards less stability because the subset of trajectories that fail hit machine precision tend to be the ones that have decayed slower. Leaving in the other part of the ensemble does not help in removing the bias either for the above stated reason that they have a higher propensity to diverge. A final possibility is that we limit the simulation period over which we consider the distances for a given τ_I to the time when the first sample reaches machine precision, a time that away from the transition is lower bounded by that given by a deterministic decay to that precision with the rate of the maximum Lyapunov exponent. However, even this introduces a bias, though in the other direction, as we likely miss many decorrelation events that fall beyond the simulation window. As a result, we falsely label as local the trajectories that were in fact from another flux tube, biasing the probability of separation to lower values.

We computed two sections just above and below the transition Figure 4.16(b). Two immedi-ately obvious features is that there are completely pixelated regions consistent with a chaotic dynamics below the transition and extended undecorrelated areas of apparent stability above the transition. While the first can be explained by the chosen resolution of the grid of initial conditions being too coarse to resolve the smaller tubes, the latter cannot be consistent with λ_max > 0 and suggest the existence of decorrelation times beyond the chosen simulation win-dow. To further understand the features in these sections, we evolved a long trajectory starting from a pair of pixels, one of each type, for both below and above the transition. The results conform to those provided by the behaviour of λ_max with τ_I. The apparent tubes above the transition are exposed as artefacts by realizing that we have just not simulated long enough to allow the instability time to decorrelate the microstate due to the aforementioned long tempo-ral correlations in the distance. Indeed, these sections were constructed from simulation times long enough to contain the decorrelation events of the corresponding LIF network. However, given the temporal correlations, perhaps this estimate is no longer valid. We computed a set

of P_s() for a range of τ_I approaching the transition and with exponentially longer simulation windows Figure 4.16(a). Unfortunately, the biases discussed in the previous section still have the expected effect on the numerically computed probability of separation curves Figure 4.16.

In particular, the curves obtain an artefactual finite limiting value for → 0. Increasing the precision from double to long double allows the numerics to get closer to the transition without exhibiting a bias, but only marginally. The effect will always skew results within some distance from the critical value.