Fig 4. History dependence dismisses redundant past dependencies and captures synergistic effects.

(A,B) Analysis of a subsampled branching process. (A) The population activity was simulated as a branching process (m = 0.98) and subsampled to yield the spike train of a single neuron (Materials and methods). (B) Autocorrelation C(T) and lagged mutual information L(T) include redundant dependencies and decay much slower than the gain ΔR(T), with much longer timescales (vertical dashed lines). (C,D) Analysis of an Izhikevich neuron in chattering mode with constant input and small voltage fluctuations. The neuron fires in regular bursts of activity. (D) Time-lagged measures C(T) and L(T) measure both, intra- (T < 10 ms) and inter-burst (T > 10 ms) dependencies, which decay very slowly due to regularity of the firing. The gain ΔR(T) reflects that most spiking can already be predicted from intra-burst dependencies, whereas inter-burst dependencies are highly redundant. In this case, only ΔR(T) yields a sensible time scale (blue dashed line). (E,F) Analysis of a generalized leaky integrate-and-fire neuron with long-lasting adaptation filter ξ [3, 43] and constant input. (F) Here, ΔR(T) decays slower to zero than the autocorrelation C(T), and is higher than L(T) for long T. Therefore, the dependence on past spikes is stronger when taking more-recent past spikes into account (ΔR(T)), as when considering them independently (L(T)). Due to these synergistic past dependencies, ΔR(T) is the only measure that captures the long-range nature of the spike adaptation.