This paper presents a fully implantable neural recording system for the simultaneous recording of 128 channels. The electrocorticography (ECoG) signals are sensed with 128 gold electrodes embedded in a 10 µm thick polyimide foil. The signals are picked up by eight amplifier array ICs and digitized with a resolution of 16 bit at 10 kHz. The digitized measurement data is processed in a reconfigurable digital ASIC, which is fabricated in a 0.35 µm CMOS technology and occupies an area of 2.8×2.8mm2. After data reduction, the measurement data is fed into a transceiver IC, which transmits the data with up to 495 kbit/s to a base station, using an RF loop antenna on a flexible PCB. The power consumption of 84mW is delivered via inductive coupling from the base station.

The speed and reliability of mammalian perception indicate that cortical computations can rely on very few action potentials per involved neuron. Together with the stochasticity of single-spike events in cortex, this appears to imply that large populations of redundant neurons are needed for rapid computations with action potentials. Here we demonstrate that very fast and precise computations can be realized also in small networks of stochastically spiking neurons. We present a generative network model for which we derive biologically plausible algorithms that perform spike-by-spike updates of the neuron's internal states and adaptation of its synaptic weights from maximizing the likelihood of the observed spike patterns. Paradigmatic computational tasks demonstrate the online performance and learning efficiency of our framework. The potential relevance of our approach as a model for cortical computation is discussed.

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 ,f *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 p(kIJL(x)) = (JL~))k exp{-JL(X)} ,

Efficient coding has been proposed as a first principle explaining neuronal response properties in the central nervous system. The shape of optimal codes, however, strongly depends on the natural limitations of the particular physical system. Here we investigate how optimal neuronal encoding strategies are influenced by the finite number of neurons N (place constraint), the limited decoding time window length T (time constraint), the maximum neuronal firing rate f(max) (power constraint), and the maximal average rate (f)(max) (energy constraint). While Fisher information provides a general lower bound for the mean squared error of unbiased signal reconstruction, its use to characterize the coding precision is limited. Analyzing simple examples, we illustrate some typical pitfalls and thereby show that Fisher information provides a valid measure for the precision of a code only if the dynamic range (f(min)T, f(max)T) is sufficiently large. In particular, we demonstrate that the optimal width of gaussian tuning curves depends on the available decoding time T. Within the broader class of unimodal tuning functions, it turns out that the shape of a Fisher-optimal coding scheme is not unique. We solve this ambiguity by taking the minimum mean square error into account, which leads to flat tuning curves. The tuning width, however, remains to be determined by energy constraints rather than by the principle of efficient coding.