Time encoding with the integrate and fire neuron

Aurel Lazar

Department of Electrical Engineering
Columbia University

A key question in theoretical neuroscience is how to represent an arbitrary stimulus as a sequence of action potentials. The temporal requirements imposed on this representation might dependent on the information presented to the sensory neurons. The temporal precision of auditory processing, for example, involves measurements of interaural time delays with sub millisecond accuracy (Hudspeth and Konishi, PNAS, 97:11690-11691). This imposes very stringent temporal requirements on the transduction process. We formulate the question of stimulus representation as one of time encoding, i.e., as one of encoding amplitude information into a time sequence. A Time Encoding Machine is the realization of such a mechanism.

We show that a Time Encoding Machine consisting of an integrate and fire neuron with feedback is invertible. Under simple conditions, bandlimited stimuli encoded with the Time Encoding Machine can be recovered loss-free from the neural spike train at its output. This result is somewhat unexpected because the Time Encoding Machine is non-linear. Less surprising is that simple non-linear algorithms provide for perfect recovery. The recovery algorithms are realized as a Time Decoding Machine.

The Time Decoding Machine helps elucidate some of the key open questions of temporal coding. We show that:

Finally, we present the relationship between time encoding and the representation of bandlimited signals in classical information theory. In the latter, uniform sampling (Shannon's sampling theorem) together with quantization of the discrete signal amplitude is the representation of choice. We will also show the relationship between time encoding, frequency modulation and asynchronous Sigma-Delta modulation.