Reconstruction of neuromorphic dynamics from a single scalar time series using variational autoencoder and neural network map

TL;DR

This paper examines the reconstruction of a family of dynamical systems with neuromorphic behavior using a single scalar time series using a model of a physiological neuron based on the Hodgkin-Huxley formalism to train a neural network that can operate as a discrete time dynamical system with one control parameter.

Abstract

This paper examines the reconstruction of a family of dynamical systems with neuromorphic behavior using a single scalar time series. A model of a physiological neuron based on the Hodgkin-Huxley formalism is considered. Single time series of one of its variables is shown to be enough to train a neural network that can operate as a discrete time dynamical system with one control parameter. The neural network system is created in two steps. First, the delay-coordinate embedding vectors are constructed form the original time series and their dimension is reduced with by means of a variational autoencoder to obtain the recovered state-space vectors. It is shown that an appropriate reduced dimension can be determined by analyzing the autoencoder training process. Second, pairs of the recovered state-space vectors at consecutive time steps supplied with a constant value playing the role of a control parameter are used to train another neural network to make it operate as a recurrent map. The regimes of thus created neural network system observed when its control parameter is varied are in very good accordance with those of the original system, though they were not explicitly presented during training.

Figures (16)

• Figure 1: Solution of the system \ref{['eq:sys']} in the regime of bursting oscillations. $k=0$ and $V_S=-36$. Other parameters see in Tab. \ref{['tab:param']}.
• Figure 2: Spiking of the system \ref{['eq:sys']} at $k=0$ and $V_S=-33$; also see Tab. \ref{['tab:param']}.
• Figure 3: Diagram of regimes of the system \ref{['eq:sys']} at $k=0$. The horizontal axis shows the change in parameter $V_S$. For each $V_S$ a solution of the system \ref{['eq:sys']} is computed and values of $V(t)$ are shown along the vertical axis with gray shapes. The darker shades indicate higher values. The plotted solutions are recorded after omitting transients and are aligned by the smallest minimum on the curve $V(t)$.
• Figure 4: Diagram of regimes of the system \ref{['eq:sys']} at $k=1$. Panels (a) and (b) illustrate the bistability. To plot each panel we compute one of two coexisting solutions in the middle of the plot at $V_S=-36$: (a) the bursts and (b) the fixed point. Then this solution is continued to the left and to the right, i.e., computation at new parameter step starts from the last point obtained at the previous step
• Figure 5: Integrated $V(t)$ and $n(t)$, panels (a) and (b), respectively, and differentiated $S(t)$, panel (c), taken form Fig. \ref{['fig:sol_sherman_ode_o_burst']}.