Analyzing the effect of an integrate and fire encoder and decoder in feedback

The human motor system has often been studied as a feedback control problem, but the integration of binary neuronal encoding in these formulations has been largely overlooked. Neuronal encoding has, however, been studied in open loop, with efforts directed towards the perfect reconstruction of the encoded signal, which requires an infinite number of spikes. Here, we study a dynamic neuronal encoding model in the finite time setting and i) provide explicit conditions for the well-posedness and stability of the encoder embedded in a feedback loop, and ii) develop a method for the reconstruction of the reference signal if we have access to a finite number of spikes as samples of the error signal in closed loop. Moreover, we show that if the reference signal is bandlimited and in the polynomially weighted L₂ space, then it can be approximately reconstructed with an error that decreases polynomially in T when we have access to all the spikes in [-T, T].