Part of Advances in Neural Information Processing Systems 15 (NIPS 2002)
Matthias Bethge, David Rotermund, Klaus Pawelzik
Here we derive optimal gain functions for minimum mean square re(cid:173) construction from neural rate responses subjected to Poisson noise. The shape of these functions strongly depends on the length T of the time window within which spikes are counted in order to estimate the under(cid:173) lying firing rate. A phase transition towards pure binary encoding occurs if the maximum mean spike count becomes smaller than approximately three provided the minimum firing rate is zero. For a particular function class, we were able to prove the existence of a second-order phase tran(cid:173) sition analytically. The critical decoding time window length obtained from the analytical derivation is in precise agreement with the numerical results. We conclude that under most circumstances relevant to informa(cid:173) tion processing in the brain, rate coding can be better ascribed to a binary (low-entropy) code than to the other extreme of rich analog coding.
1 Optimal neuronal gain functions for short decoding time windows
The use of action potentials (spikes) as a means of communication is the striking feature of neurons in the central nervous system. Since the discovery by Adrian [1] that action poten(cid:173) tials are generated by sensory neurons with a frequency that is substantially determined by the stimulus, the idea of rate coding has become a prevalent paradigm in neuroscience [2]. In particular, today the coding properties of many neurons from various areas in the cortex have been characterized by tuning curves, which describe the average firing rate response as a function of certain stimulus parameters. This way of description is closely related to the idea of analog coding, which constitutes the basis for many neural network models. Reliablv inference from the observed number of spikes about the underlying firing rate of a neuronal response, however, requires a sufficiently long time interval, while integration times of neurons in vivo [3] as well as reaction times of humans or animals when per(cid:173) forming classification tasks [4, 5] are known to be rather short. Therefore, it is important to understand, how neural rate coding is affected by a limited time window available for decoding.
While rate codes are usually characterized by tuning functions relating the intensity of the
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*http://www.neuro.urn-bremen.dermbethge
neuronal response to a particular stimulus parameter, the question, how relevant the idea of analog coding actually is does not depend on the particular entity represented by a neuron. Instead it suffices to determine the shape of the gain function, which displays the mean fir(cid:173) ing rate as a function of the actual analog signal to be sent to subsequent neurons. Here we seek for optimal gain functions that minimize the minimum average squared reconstruction error for a uniform source signal transmitted through a Poisson channel as a function of the maximum mean number of spikes.
In formal terms, the issue is to optimally encode a real random variable x in the number of pulses emitted by a neuron within a certain time window. Thereby, x stands for the intended analog output of the neuron that shall be signaled to subsequent neurons. The latter, however, can only observe a number of spikes k integrated within a time interval of length T. The statistical dependency between x and k is specified by the assumption of Poisson noise
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