The computational power of formal models for networks of spiking neurons is compared with that of other neural network models based on McCulloch Pitts neurons (i.e. threshold gates) respectively sigmoidal gates. In particular it is shown that networks of spiking neurons are computationally more powerful than these other neural network models. A concrete biologically relevant function is exhibited which can be computed by a single spiking neuron (for biologically reasonable values of its parameters), but which requires hundreds of hidden units on a sigmoidal neural net. This article does not assume prior knowledge about spiking neurons, and it contains an extensive list of references to the currently available literature on computations in networks of spiking neurons and relevant results from neurobiology. 1 Definitions and Motivations If one classifies neural network models according to their computational units, one can distinguish three different generations. The first generation i...

An integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate-and-fire type is discussed. It is analytically shown that transients from a state of incoherent firing can be immediate. The stability of incoherent firing is analyzed in terms of the noise level and transmission delay and a bifurcation diagram is derived. The response of a population of noisy integrate-and-fire neurons to an input current of small amplitude is calculated and characterized by a linear filter L. The stability of perfectly synchronized `locked' solutions is analyzed.

We investigate the computational power of a formal model for networks of spiking neurons. It is shown that simple operations on phasedifferences between spike-trains provide a very powerful computational tool that can in principle be used to carry out highly complex computations on a small network of spiking neurons. We construct networks of spiking neurons that simulate arbitrary threshold circuits, Turing machines, and a certain type of random access machines with real valued inputs. We also show that relatively weak basic assumptions about the response- and threshold-functions of the spiking neurons are sufficient in order to employ them for such computations. 1 Introduction and Basic Definitions There exists substantial evidence that timing phenomena such as temporal differences between spikes and frequencies of oscillating subsystems are integral parts of various information processing mechanisms in biological neural systems (for a survey and references see e.g. Kandel et al., ...

Present and permanent address: Physik-Department der TU Munchen Exploiting local stability we show what neuronal characteristics are essential to ensure that coherent oscillations are asymptotically stable in a spatially homogeneous network of spiking neurons. Under standard conditions, a necessary and in the limit of a large number of interacting neighbors also sufficient condition is that the postsynaptic potential is increasing in time as the neurons fire. If the postsynaptic potential is decreasing, oscillations are bound to be unstable. This is a kind of locking theorem and boils down to a subtle interplay of axonal delays, postsynaptic potentials, and refractory behavior. The theorem also allows for mixtures of excitatory and inhibitory interactions. On the basis of the locking theorem we present a simple geometric method to verify existence and local stability of a coherent oscillation. 2 1

A theoretical model for analog computation in networks of spiking neurons with temporal coding is introduced and tested through simulations in GENESIS. It turns out that the use of multiple synapses yields very noise robust mechanisms for analog computations via the timing of single spikes in networks of detailed compartmental neuron models. One arrives in this way at a method for emulating arbitrary Hopfield nets with spiking neurons in temporal coding, yielding new models for associative recall of spatio-temporal firing patterns. We also show that it suffices to store these patterns in the efficacies of excitatory synapses. A corresponding layered architecture yields a refinement of the synfire-chain model that can assume a fairly large set of different stable firing patterns for different inputs.

2 Abstract We investigate the computational power of a formal model for networks of spiking neurons. It is shown that simple operations on phase-differences between spike-trains provide a very powerful computational tool that can in principle be used to carry out highly complex computations on a small network of spiking neurons. We construct networks of spiking neurons that simulate arbitrary threshold circuits, Turing machines, and a certain type of random access machines with real valued inputs. We also show that relatively weak basic assumptions about the response- and threshold-functions of the spiking neurons are sufficient in order to employ them for such computations. Furthermore we prove upper bounds for the computational power of networks of spiking neurons with arbitrary piecewise linear responseand threshold-functions, and show that they are with regard to realtime simulations computationally equivalent to a certain type of random access machine, and to recurrent analog neural nets with piecewise linear activation functions. In addition we give corresponding results for networks of spiking neurons with a limited timing precision, and we prove upper and lower bounds for the VC-dimension and pseudo-dimension of networks of spiking neurons. 3 1

Scene analysis in the mammalian visual system, conceived as a distributed and parallel process, faces the so-called binding problem. As a possible solution, the temporal correlation hypothesis has been suggested and implemented in phase-coding models.

This paper discusses the relation of theory and experiment in neuroscience exemplified by three assumptions often made in models of coherent activation in the cortex: basic feature-coding oscillators, phase-coding and global binding of whole objects. Apparently these assumptions are not very well supported by the experimental evidence. We propose that it is the single synchronized population-burst that matters: spikes of feature-coding cells are temporally clustered in our opinion by recurrent associative processes. In each burst a single stimulus is processed (if there are several). Synchronization is restricted to cortical sites which physically interact. These principles are illustrated by computer simulations.

How does a neuron vary its mean output firing rate if the input changes from random to coherent activity? What are the critical parameters of the neuronal dynamics and input statistics? To answer these questions, we investigate the coincidence detection properties of an integrate-and-fire neuron. We derive an expression indicating how coincidence detection depends on neuronal parameters. Specifically, (i) we show how coincidence detection depends on the shape of the postsynaptic response function, the number of synapses, and the input statistics, and (ii) we demonstrate that there is an optimal threshold. Our considerations can be used to predict from neuronal parameters whether and to what extent a neuron can act as a coincidence detector and thus can convert a temporal code into a rate code. Physik-Department der TU Munchen (T35), D-85747 Garching bei Munchen, Germany y Swiss Federal Institute of Technology, Center of Neuromimetic Systems, EPFL-DI, CH-1015 Lausanne, Switz...

. We present the requirements for a neurocomputer for spike-processing neural networks. In a simulation study we investigated the performance of available hardware and showed, that there is still a need for a specific neurocomputer dedicated to the simulation of spike-processing networks. On the basis of our simulation study and an investigation of the features of spike-processing networks we analyses the requirements for the design of dedicated hardware. An efficient hardware architecture should contain an event-list module, a sender-oriented connection module and a number of fixed-point processing units. 1 Introduction Experimental results [1] [2] together with theoretical studies [3] [4] suggest that the time structure of neuronal spike trains is relevant in neuronal signal processing. The synchronized firing of neuronal assemblies could serve as a versatile and general mechanism for feature binding, pattern segmentation and figure/ground separation. This mechanism could also be u...