Reconstruction of a neural network from a time series of firing rates

Randomly coupled neural fields demonstrate irregular variation of firing rates, if the coupling is strong enough, as has been shown by [Phys. Rev. Lett. 61, 259 (1988)]. We present a method for reconstruction of the coupling matrix from a time series of irregular firing rates. The approach is based on the particular property of the nonlinearity in the coupling, as the latter is determined by a sigmoidal gain function. We demonstrate that for a large enough data set and a small measurement noise, the method gives an accurate estimation of the coupling matrix and of other parameters of the system, including the gain function.

Figure 1

Example of chaotic neural fields (first 20 fields $x_i\left(t\right)$, for $i=1,...,20$ are depicted with different colors).

Figure 2

Solid red line: $y\left(t\right)={\tau }_1{\dot{x}}_1+x_1$ is sampled with time step $0.05$, points where $|\dot{y}|>0.3$ are shown with blue circles. One can see that selection of relatively rapidly varying parts of the time series allows one to avoid nonsensitive epochs where the field $y\left(t\right)$ is nearly constant.

Figure 3

Dependence of the minimal singular value on the parameter $\tau$ for different lengths of time series (from top to bottom: total used time intervals $2500,1250,500,250,100$). The vertical line shows the true value of $\tau$.

Figure 4

Original coupling constants $w_{1j}$ (circles) and the reconstructed ones $w_{1j}^r$ for the data sets with total used time intervals $T=2500,1250,500,250$ (the corresponding markers). In these sets the number of data points used for reconstruction was $8961,4174,1627,796$, respectively.

Figure 5

Median errors for the reconstruction depicted in Fig. 4, as functions of the total time interval used.

Figure 6

Reconstruction of the gain function $F_1$. The same data points as in finding the coupling matrix Fig. 4, with $T=2500$, are used. Points depict values of ${\tau }_1{\dot{x}}_1+x_1$ versus ${\sum }_k{w}_{1k}{x}_k$ at all recorded time instants, where $w_{1k}$ are the reconstructed coupling constants.

Figure 7

Original coupling constants $w_{1j}$ (circles) and the reconstructed ones $w_{1j}^r$ for the data sets with total used time interval $T=5000$, for two values of noise intensity: $\delta =0.012$ and $\delta =0.016$. The noisy data sets have been preprocessed with the Savitzky-Golay filter of order (16, 16, 4) (see Ref. [14]), the same filter has been used to calculate the derivatives.