  We investigated a model for the neural integrator based on hysteretic units
connected by positive feedback. Hysteresis is assumed to emerge from the
intrinsic properties of the cells. We consider the recurrent networks
containing either bistable or multistable neurons. We apply our analysis to the
oculomotor velocity-to-position neural integrator that calculates the eye
positions from the inputs that carry information about eye angular velocity.
Using the analysis of the system in the parameter space we show the following.
The direction of hysteresis in the neuronal response may be reversed for the
system with recurrent connections compared to the case of unconnected neurons.
Thus, for the NMDA receptor based bistability the firing rates after ON
saccades may be higher than after OFF saccades for the same eye position. We
suggest that this is an emergent property due to the presence of global
recurrent feedback. The reversal of hysteresis occurs only when the size of
hysteresis differs from neuron to neuron. We also relate the macroscopic leak
time-constant of the integrator to the rate of microscopic spontaneous
noise-driven transitions in the hysteretic units. Finally, we argue that the
presence of neurons with small hysteresis may remove the threshold for
integration.
