J. L. van Hemmen

Associative memory in a network of ‘spiking’ neurons

The Hopfield network provides a simple model of an associative memory in a neuronal structure. It is, however, based on highly artificial assumptions, especially the use of formal two-state neurons or graded-response neurons. The authors address the question of what happens if formal neurons are replaced by a model of ‘spiking’ neurons. They do so in two steps. First, they show how to include refractoriness and noise into a simple threshold model of neuronal spiking. The spike trains resulting from such a model reproduce the distribution of interspike intervals and gain functions found in real neurons. In a second step they connect the model neurons so as to form a large associative memory system. The spike transmission is described by a synaptic kernel which includes axonal delays, ‘Hebbian’ synaptic efficacies, and a realistic postsynaptic response. The collective behaviour of the system is predicted by a set of dynamical equations which are exact in the limit of a large and fully connected network that...

Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators

A Lyapunov function for the phase-locked state of the Kuramoto model of non-linearly coupled oscillators is presented. It is also valid for finite-range interactions and allows the introduction of thermodynamic formalism such as ground states and universality classes. For the Kuramoto model, a minimum of the Lyapunov function corresponds to a ground state of a system with frustration: the interaction between the oscillators,XY spins, is ferromagnetic, whereas the random frequencies induce random fields which try to break the ferromagnetic order, i.e., global phase locking. The ensuing arguments imply asymptotic stability of the phase-locked state (up to degeneracy) and hold for any probability distribution of the frequencies. Special attention is given to discrete distribution functions. We argue that in this case a perfect locking on each of the sublattices which correspond to the frequencies results, but that a partial locking of some but not all sublattices is not to be expected. The order parameter of the phase-locked state is shown to have a strictly positive lower bound (r ⩾ 1/2), so that a continuous transition to a nonlocked state with vanishing order parameter is to be excluded.

Escape over a fluctuating barrier: The white noise limit

The authors examine the problem of diffusion over a fluctuating barrier in the limit where the barrier fluctuations are extremely fast compared with all other timescales in the problem. In the white noise limit, the decay of probability from the metastable state is exponential with a characteristic timescale exhibiting Arrhenius behaviour. For small but finite correlation times for the fluctuating part of the potential, the effective barrier increases with the correlation time. Implications for liquids above the glass transition are discussed.

Tunnelling of Quantum Spins

A WKB formalism is presented to describe the quantum dynamics, including tunnelling, of a single spin with large spin quantum number. This is of particular relevance to the treatment of Hamiltonians with a large anisotropy. For a wide class of Hamiltonians we obtain the eigenstates and tunnelling rates and show the universal dependence of the latter upon the spin quantum number, anisotropy and transverse fields.

The Thermodynamic Limit for Long-Range Random Systems

Long-range spin systems with random interactions are considered. A simple argument is presented showing that the thermodynamic limit of the free energy exists and depends neither on the specific random configuration nor on the sample shape, provided there is no external field. The argument is valid for both classical and quantum spin systems, and can be applied to (a) spins randomly distributed on a lattice and interacting via dipolar interactions; and (b) spin systems with potentials of the formJ(x1,x2)/|x1 -x2|αd, where theJ(x1,x2) are independent random variables with mean zero,d is the dimension, and α > 1/2. The key to the proof is a (multidimensional) subadditive ergodic theorem. As a corollary we show that, for random ferromagnets, the correlation length is a nonrandom quantity.

The Hebb Rule: Storing Static and Dynamic Objects in an Associative Neural Network

The Hebb rule (Hebb, 1949) indicates how information presented to a neural network during a learning session is stored in the synapses, local elements which act as mediators between neurons. In this paper we demonstrate that the Hebb rule can be used to handle both stationary and dynamic objects such as single patterns and cycles. The two main ideas are: a) a broad distribution of delays as they occur in the natural dynamics and b) incorporation of the very same delays during the learning session. Our work shows that the resulting procedure is robust and faithful.

Mean exit times over fluctuating barriers

Abstract We investigate the problem of thermal activation over a fluctuating barrier. Three regimes are considered: the fluctuations slow compared to the mean crossing time τA of the average barrier height, fluctuations on roughly the same timescale as τA, and fluctuations extremely fast compared to τA. In the latter two cases, the mean barrier crossing time is reduced. The relevance of these results to a variety of problems in complex systems is discussed.

Spike-timing-dependent plasticity for neurons with recurrent connections

The dynamics of the learning equation, which describes the evolution of the synaptic weights, is derived in the situation where the network contains recurrent connections. The derivation is carried out for the Poisson neuron model. The spiking-rates of the recurrently connected neurons and their cross-correlations are determined self- consistently as a function of the external synaptic inputs. The solution of the learning equation is illustrated by the analysis of the particular case in which there is no external synaptic input. The general learning equation and the fixed-point structure of its solutions is discussed.

DEVELOPMENT OF SPATIOTEMPORAL RECEPTIVE FIELDS OF SIMPLE CELLS : I. MODEL FORMULATION

Abstract. A model for the development of spatiotemporal receptive fields of simple cells in the visual cortex is proposed. The model is based on the 1990 hypothesis of Saul and Humphrey that the convergence of four types of input onto a cortical cell, viz. non-lagged ON and OFF inputs and lagged ON and OFF inputs, underlies the spatial and temporal structure of the receptive fields. It therefore explains both orientation and direction selectivity of simple cells. The response properties of the four types of input are described by the product of linear spatial and temporal response functions. Extending the 1994 model of one of the authors (K.D. Miller), we describe the development of spatiotemporal receptive fields as a Hebbian learning process taking into account not only spatial but also temporal correlations between the different inputs. We derive the correlation functions that drive the development both for the period before and after eye-opening and demonstrate how the joint development of orientation and direction selectivity can be understood in the framework of correlation-based learning. Our investigation is split into two parts that are presented in two papers. In the first, the model for the response properties and for the development of direction-selective receptive fields is presented. In the second paper we present simulation results that are compared with experimental data, and also provide a first analysis of our model.

Tunneling of quantum spins

Abstract A WKB formalism is presented whereby the tunneling rate of a quantum spin is obtained in the semiclassical limit when h → 0 and the spin quantum number S → ∞ in such a way that h S remains constant. The main idea is to single out one of t anisotropy axes, say the z-axis, to work in a representation with Sz, the z-component of the spin, diagonal and to describe quantum tunneling as a hopping process on the spectrum of Sz. This formalism enables us to efficiently handle tunneling problems, to incorporate dissipation, and to prove that the tunneling rate is universal, i.e. independent of the particular form of the anisotropy.