Spike-based synaptic plasticity and the emergence of direction selective simple cells: mathematical analysis

In the companion paper we presented extended simulations showing that the recently observed spike-timing dependent synaptic plasticity can explain the development of simple cell direction selectivity (DS) when simultaneously modifying the synaptic strength and the degree of synaptic depression. Here we estimate the spatial shift of the simple cell receptive field (RF) induced by the long-term synaptic plasticity, and the temporal phase advance caused by the short-term synaptic depression in response to drifting grating stimuli. The analytical expressions for this spatial shift and temporal phase advance lead to a qualitative reproduction of the frequency tuning curves of non-directional and directional simple cells. In agreement with in vivo recordings, the acquired DS is strongest for test gratings with a temporal frequency around 1-4 Hz. In our model this best frequency is determined by the width of the learning function and the time course of depression, but not by the temporal frequency of the 'training' stimuli. The analysis further reveals the instability of the initially symmetric RF, and formally explains why direction selectivity develops from a non-directional cell in a natural, directionally unbiased stimulation scenario.