Variable Action Potential Onset Rapidness
3.6 Entangled Statistics
the standard deviation goes to zero asσρ ∼1/√
N, indicating broadly distributed pairwise corre-lations. One might speculate whether this fact of a broad distribution might imply higher order correlations. Another interesting observation is the increase of the width of the pairwise correla-tions with the square root of the network-averaged firing rate σρ ∼√
ν¯. In excitatory-inhibitory networks, the activation of the excitatory-inhibitory feedback loops with ε slightly increased the average correlations but did not change the width of the distributions. Altogether, this analysis of the pairwise spike correlations confirmed the prediction of weak and broadly distributed pairwise spike correlations in the balanced state.
3.6 Entangled Statistics
Weak pairwise correlations do not necessarily imply vanishing higher order correlations (see Sec-tion 1.5). Can we deduce more about the strength of higher order correlaSec-tion from our analysis?
We have shown that rapid theta neuron networks undergo a transition from chaotic to stable dynam-ics at a parameter dependent critical AP onset rapidness rc. Networks with chaotic dynamics for r<rcexhibit strange attractors with fractal dimensionsD>1. Networks with stable dynamics for r>rc exhibit one-dimensional attractorsD=1 which implies periodic orbits. One would expect that lower dimensional attractors imply strongly correlated network states, but the pairwise spike correlations were generally weak independent of this dramatic change of the collective networks dynamics.
In fact, the attractor dimension can be related to the highest ordernof nonvanishing correlations betweenn neuronsC(n) =hsi(t)sj(t0). . .i. The first order correlations describe the average firing rates of the neurons, the second order the pairwise spike correlations and so on. In a network of N independent or non-interacting neurons, the attractor dimension would beD=N. The indepen-dence means that there are no higher order correlations, thus n=1. In a stable periodic network state, the attractor dimension isD=1 and all neurons would be statistically dependent, thusn=N.
If the network state was quasiperiodic with xincommensurable frequencies, the attractor dimen-sion would beD=xand correlations of ordern≥N−x+1 would vanish. We can thus conjecture that generally correlations of higher order thann=N−D+1 vanish.
The decreasing attractor dimension for increasing AP onset rapidness (Fig. 3.13) thus implies that the neurons become more entangled for larger AP onset rapidness despite weak pairwise correlations. We will demonstrate this in an example of a small network for different values of AP onset rapidness r (Fig. 3.24). Displayed are the spike trains of 10 out of 20 neurons and the sections of the phases of neuron 1 and 2 at times when neuron 3 spikes. The Poincaré-sections represent a cut through the attractor. All spike trains show the asynchronous and irregular firing activity in these networks. Below the critical AP onset rapidness rc =12, the Poincaré-sections are a chaotic cloud. The dimensionality of the attractor is D≈12.5 for r=1 and stays
D&3 up to the transition to stable dynamics atrc=12. Withr=12, one can see that the firing
pattern of the entire network becomes periodic as expected in the regime of stable dynamics. The Poincaré- sections for r=12 and r=100 display a cut through the one-dimensional attractors, thus dots. The number of dots represents the period length which is larger than one. We can thus conclude that despite weak pairwise spike correlations and asynchronous firing of the neurons, the statistics or neural networks becomes more entangled with higher AP onset rapidness.
The period length indicated by the number of dots in the Poincaré-sections forr>rcincreased exponentially with the number of neurons, together with the transient lengths to the periodic states.
This characteristic of balanced networks with very fast AP onset rapidness is similar to leaky
0 t (s) 4 1
5 10
i
r = 1 (D=12.5)
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1 5 10
r = 11 (D=3.3)
0 t (s) 4
1 5 10
r = 12 (D=1)
0 t (s) 4
1 5 10
r = 100 (D=1)
-π 0 π
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φ 2
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Figure 3.24 – Entangled statistics in rapid theta neuron networks. Upper panels spike trains of 10 randomly chosen neurons, lower panels: Poincaré-sections of the phases of neurons 1 and 2 when neuron 3 emits a spike, the AP onset rapidnessrwas increased from left to right in otherwise identical networks, (parameters:N=20,K=10, ¯ν=10 Hz,J0=1,τm=10 ms; data from 100 spikes per neuron on average).
integrate and fire networks and should be investigated further for a thorough understanding of the balanced state in the stable regime.
3.7 Summary
3.7 Summary
In this chapter we investigated the influence of the action potential (AP) onset rapidness of the individual neurons on the collective network dynamics. A new exactly solvable neuron model with variable AP onset rapidness, called the rapid theta neuron model, allowed for the direct application of the approach to study neural network dynamics introduced in Chapter 2. The presented results show that the AP onset rapidness strongly affects the collective network dynamics.
Networks of rapid theta neurons in the balanced state undergo a phase transition from chaotic to stable dynamics at a parameter-dependent critical AP onset rapidnessrc. We call this the edge of chaos. The general properties of the balanced state such as the macroscopic firing statistics and the weak pairwise spike correlations are basically independent of the AP onset rapidness and fail to capture the dramatic change of the underlying network dynamics.
The AP onset rapidness also influences the dynamics within the chaotic regime qualitatively.
We have observed two different dynamic regimes indicated by a peak in the largest Lyapunov exponent with respect to the AP onset rapidnessr. For lowr<10, the network dynamics is very similar to the dynamics of theta neurons (r=1) studied in Chapter 2. For larger>10, the network dynamics shows qualitative differences to the dynamics of theta neurons. Both phase transitions to a synchronous state disappear and the intensity of the chaos decreases leading to the edge of chaos and stable dynamics for very large r. Because we expect cortical neurons to exhibit such a large AP onset rapidness, we can conclude that the collective dynamics of cortical neurons qualitatively differs from the collective dynamics of neurons with low AP onset rapidness.
The network dynamics of rapid theta neurons is characterized by:
• Deterministic chaos, characterized by positive and finite Lyapunov exponents. A peak in the largest Lyapunov exponent for increasing AP onset rapidness indicates a qualitative change of the network dynamics.
• Extensive chaos, characterized by network size-invariant Lyapunov spectra and a linear in-crease of the number of positive Lyapunov exponents, attractor dimension and entropy pro-duction rate with the number of neurons.
• Monotonously decreasing attractor dimension with increasing AP onset rapidness, implying entangled statistics towards the edge of chaos despite generally weak pairwise spike corre-lations in the balanced state.
• Monotonously decreasing entropy production rate with increasing AP onset rapidness, im-plying reduced loss of information in cortical networks through larger AP onset rapidness of the single neurons.
For very large AP onset rapidness, the AP initiation becomes basically instantaneous similar to the leaky integrate and fire model. Our results demonstrate that the characterization of the dynamics of inhibitory networks in the limit of instantaneous AP initiation (r→∞) depends on the order in which this limit and the large system limit are taken. It can either be characterized as extremely chaotic (λmax =∞) or stable (λmax = 0). Such inhibitory networks of leaky integrate and fire neurons were previously shown to exhibit stable chaos [4–7]. How stable this form of irregular dynamics is with respect to finite perturbations and temporally extended synaptic transmission will be the topic of the next two chapters.