Time encoding with the integrate and fire neuron
Department of Electrical Engineering
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.
Stimuli encoded by a single integrate and fire neuron can be recovered
loss-free from the neural spike train. The recovery of the stimulus from a
single running experiment is a defining biological requirement.
The error introduced by dropping individual action potentials or by
measurement jitter of the time of occurrence of action potentials can be
explicitly evaluated. The resulting error is a measure that quantifies the
importance of temporal coding.
Stimulus time-delays in the sub millisecond range can be readily