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., ...

Machines and Recursive Definitions 2.1 Abstract Machines The best-known model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental Church-Turing Thesis : A function f : N # N on the natural numbers (or, more generally, on strings from a finite alphabet) is computable in principle exactly when it can be computed by a Turing Machine. The Church-Turing Thesis grounds proofs of undecidability and it is essential for the most important applications of logic. On the other hand, it cannot be argued seriously that Turing machines model faithfully all algorithms on the natural numbers. If, for example, we code the input n in binary (rather than unary) notation, then the time needed for the computation of f(n) can sometimes be considerably shortened; and if we let the machine use two tapes rather than one, then (in some cases) we may gain a quadratic speedup of the computation, see [8]. This mea...

machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the well-known definition (e.g., see [6]), 13 we recall that each Turing machine M is equipped with a "semi-infinite tape" which it uses both to compute and also to communicate with its environment: To determine the value f(n) (if any) of the partial function 14 f : N * N computed by M , we put n on the tape in some standard way, e.g., by placing n + 1 consecutive 1s at its beginning; we start the machine in some specified, initial, internal state q 0 and looking at the leftmost end of the tape; and we wait until the machine stops (if it does), at which time the value f(n) can be read off the tape, by counting the successive 1s at the left end. Turing argued that the number-theoretic functions which can (in principle) be computed by any deterministic, physical device are exactly those which can be computed by a Turing machine, and the correspon...

. We show that a Turing machine with two single-head one-dimensional tapes cannot recognize the set f x2x 0 j x 2 f0; 1g and x 0 is a prefix of x g in real time, although it can do so with three tapes, two two-dimensional tapes, or one two-head tape, or in linear time with just one tape. In particular, this settles the longstanding conjecture that a two-head Turing machine can recognize more languages in real time if its heads are on the same one-dimensional tape than if they are on separate one-dimensional tapes. 1. Introduction The Turing machines commonly used and studied in computer science have separate tapes for input/output and for storage, so that we can conveniently study both storage as a dynamic resource and the more complex storage structures required for efficient implementation of practical algorithms [HS65]. Early researchers [MRF67] asked specifically whether two-head storage is more powerful if both heads are on the same one-dimensional storage tape than if t...

In this paper we show that single-tape nondeterministic time T (n) bounded Turing Machines can be simulated on single-tape nondeterministic p T (n) space-bounded Turing Machines in time T (n). 0.1 Introduction Determining the relationship between recognizing power of resource-bounded machines is a central and long standing open problem in Computational Complexity Theory. The study of different machine models for sequential computation is essential to obtain a model-independent complexity theory, or to translate results obtained for one model directly to other models without proving them again. Many theorems have been proved on the relation between different variants of the Turing Machine model. For a Turing Machine (TM for short) time is measured by the number of transitions in a computation and space is measured by the number of different tape cells visited by the head(s) during the computation As far as space is concerned all different models can simulate each other with constan...

machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the well-known definition (e.g., see [6]), we recall that each Turing machine M is equipped with a "semi-infinite tape" which it uses both to compute and also to communicate with its environment: To determine the value f(n) (if any) of the partial function f : N * N computed by M , we put n on the tape in some standard way, e.g., by placing n + 1 consecutive 1s at its beginning; we start the machine in some specified, initial, internal state q 0 and looking at the leftmost end of the tape; and we wait until the machine stops (if it does), at which time the value f(n) can be read off the tape, by counting the successive 1s at the left end. Turing argued that the number-theoretic functions which can (in principle) be computed by any deterministic, physical device are exactly those which can be computed by a Turing machine, and the corresponding version of this claim for partial functions has come to be known as the ChurchTuring Thesis, because an equivalent claim was made by Church at about the same time. Turing's brilliant analysis of "mechanical computation" in [20] and a huge body of work in the last sixty years has established the truth of the Church-Turing Thesis beyond reasonable doubt; it is of immense importance in the derivation of foundationally significant undecidability results from technical theorems about Turing machines, and it has been called "the first natural law of pure mathematics." Turing machines capture the notion of mechanical computability of numbertheoretic functions, by the Church-Turing Thesis, but they do not model faith- It has also been suggested that we do not need algorithms, only the equival...