Analysis of Spike Statistics in Neuronal Systems with Continuous Attractors or Multiple, Discrete Attractor States
Attractor networks are likely to underly working memory and integrator circuits in the brain. It is unknown whether continuous quantities are stored in an analog manner, or discretized and stored in a set of discrete attractors. In order to investigate the important issue of how one can differentiate the two systems, here we compare the neuronal spiking activity that arises from a continuous (line) attractor with that from a series of discrete attractors. Stochastic fluctuations cause the
... ons cause the position of the system along its continuous attractor to vary as a random walk. Whereas in a discrete attractor, noise causes spontaneous transitions to occur between discrete states at random intervals. We calculate the statistics of spike trains of neurons firing as a Poisson process with rates that vary according to the underlying attractor network. Since individual neurons fire spikes probabilistically, and since the state of the network as a whole drifts randomly, the spike trains of individual neurons follow a doubly stochastic (Poisson) point process. We compare the series of spike trains from the two systems using the autocorrelation function, Fano factor and inter-spike interval (ISI) distribution. Although the variation in rate can be dramatically different, especially for short time intervals, surprisingly both the autocorrelation functions and Fano factors are identical, given appropriate scaling of the noise terms. Since the range of firing rates is limited in neurons, we also investigate systems for which the variation in rate is bounded by either rigid limits, or because of leak to a single attractor state, such as the Ornstein-Uhlenbeck process. In these cases, the time-dependence of the variance in rate can be different between discrete and continuous systems, allowing in principle for these processes to be distinguished using second-order spike statistics.