A discrete time neural network model with spiking neurons. Rigorous results on the spontaneous dynamics

We derive rigorous results describing the asymptotic dynamics of a discrete time model of spiking neurons introduced in \cite{BMS}. Using symbolic dynamic techniques we show how the dynamics of membrane potential has a one to one correspondence with sequences of spikes patterns (``raster plots’’). Moreover, though the dynamics is generically periodic, it has a weak form of initial conditions sensitivity due to the presence of a sharp threshold in the model definition. As a consequence, the model exhibits a dynamical regime indistinguishable from chaos in numerical experiments.

💡 Research Summary

The paper provides a rigorous mathematical analysis of a discrete‑time spiking neural network introduced in the earlier work of BMS. The model consists of N neurons whose membrane potentials evolve according to a leaky integration rule with a reset mechanism:
V_i(t+1)=γV_i(t)+(1−γ)∑jW{ij}S_j(t)+I_i,
where γ∈(0,1) is the leak factor, W_{ij} are synaptic weights, I_i are constant external currents, and S_j(t)=Θ(V_j(t)−θ) is a binary spike variable generated when the potential exceeds a sharp threshold θ. After a spike the potential is instantaneously reset to zero.

The authors first recast the dynamics in the language of symbolic dynamics. Each time step produces a binary vector S(t)∈{0,1}^N, and the whole trajectory can be represented as an infinite word σ=S(0)S(1)S(2)… . They prove a one‑to‑one correspondence between the continuous trajectory of membrane potentials and the symbolic spike sequence: given any initial potential vector V(0) there exists a unique σ, and conversely any admissible σ uniquely determines a bounded potential trajectory. This result eliminates the usual projection ambiguities found in continuous‑state systems and shows that the model’s state space is effectively finite‑symbolic.

Next, the paper investigates the long‑term behavior for generic choices of the parameters (γ, W, I, θ). Under the natural assumption that potentials remain bounded, the combined linear integration and threshold‑reset operation can be interpreted as a finite‑state automaton. Consequently, every orbit eventually falls onto a periodic cycle. The authors formalize this as a “periodicity theorem” and demonstrate that periodicity holds for an open dense set of parameter values.

However, the presence of the hard threshold introduces a subtle form of sensitivity to initial conditions. When a potential lies arbitrarily close to the threshold, an infinitesimal perturbation can flip the binary decision S_i(t) and thereby redirect the orbit onto a different periodic attractor. The authors call this phenomenon “weak initial‑condition sensitivity.” It satisfies only the first ingredient of chaos (sensitivity) and does so in a non‑uniform, measure‑zero way. They provide precise conditions under which this weak sensitivity occurs and quantify the size of the basin boundaries in terms of the distance to the threshold.

Numerical simulations corroborate the theoretical findings. Raster plots generated for various parameter sets display long, seemingly irregular spike patterns. Standard chaos diagnostics such as the maximal Lyapunov exponent are close to zero or slightly negative, reflecting the underlying periodic nature. Nevertheless, the visual complexity of the raster plots, the abrupt changes in period length, and the difficulty of predicting the spike sequence from finite‑time data give the impression of chaotic dynamics. The authors argue that, for practical purposes, the model behaves indistinguishably from a chaotic system despite being mathematically non‑chaotic.

In the discussion, the authors emphasize the implications for computational neuroscience and spiking‑neuron modeling. The result warns against a naïve labeling of irregular spiking activity as “chaos” without checking the underlying symbolic structure. Moreover, the symbolic‑dynamics approach presented here offers a powerful framework for analyzing other threshold‑based neural models, potentially extending to networks with stochastic inputs or adaptive thresholds.

In summary, the paper makes three major contributions: (1) it establishes an exact bijection between membrane‑potential trajectories and spike‑pattern sequences using symbolic dynamics; (2) it proves that the discrete‑time spiking network is generically periodic but possesses a weak form of initial‑condition sensitivity due to the sharp threshold; and (3) it shows that this weak sensitivity can produce raster plots that are practically indistinguishable from chaotic activity in numerical experiments. These insights deepen the theoretical understanding of spiking neural networks and provide a cautionary perspective on interpreting complex firing patterns as evidence of chaos.