Artificial Neurons with Arbitrarily Complex Internal Structures

Artificial neurons with arbitrarily complex internal structure are introduced. The neurons can be described in terms of a set of internal variables, a set activation functions which describe the time evolution of these variables and a set of characteristic functions which control how the neurons interact with one another. The information capacity of attractor networks composed of these generalized neurons is shown to reach the maximum allowed bound. A simple example taken from the domain of pattern recognition demonstrates the increased computational power of these neurons. Furthermore, a specific class of generalized neurons gives rise to a simple transformation relating attractor networks of generalized neurons to standard three layer feed-forward networks. Given this correspondence, we conjecture that the maximum information capacity of a three layer feed-forward network is 2 bits per weight.Comment: 22 pages, 2 figure

A three-threshold learning rule approaches the maximal capacity of recurrent neural networks

Understanding the theoretical foundations of how memories are encoded and retrieved in neural populations is a central challenge in neuroscience. A popular theoretical scenario for modeling memory function is the attractor neural network scenario, whose prototype is the Hopfield model. The model has a poor storage capacity, compared with the capacity achieved with perceptron learning algorithms. Here, by transforming the perceptron learning rule, we present an online learning rule for a recurrent neural network that achieves near-maximal storage capacity without an explicit supervisory error signal, relying only upon locally accessible information. The fully-connected network consists of excitatory binary neurons with plastic recurrent connections and non-plastic inhibitory feedback stabilizing the network dynamics; the memory patterns are presented online as strong afferent currents, producing a bimodal distribution for the neuron synaptic inputs. Synapses corresponding to active inputs are modified as a function of the value of the local fields with respect to three thresholds. Above the highest threshold, and below the lowest threshold, no plasticity occurs. In between these two thresholds, potentiation/depression occurs when the local field is above/below an intermediate threshold. We simulated and analyzed a network of binary neurons implementing this rule and measured its storage capacity for different sizes of the basins of attraction. The storage capacity obtained through numerical simulations is shown to be close to the value predicted by analytical calculations. We also measured the dependence of capacity on the strength of external inputs. Finally, we quantified the statistics of the resulting synaptic connectivity matrix, and found that both the fraction of zero weight synapses and the degree of symmetry of the weight matrix increase with the number of stored patterns.Comment: 24 pages, 10 figures, to be published in PLOS Computational Biolog

Attractor Metadynamics in Adapting Neural Networks

Slow adaption processes, like synaptic and intrinsic plasticity, abound in the brain and shape the landscape for the neural dynamics occurring on substantially faster timescales. At any given time the network is characterized by a set of internal parameters, which are adapting continuously, albeit slowly. This set of parameters defines the number and the location of the respective adiabatic attractors. The slow evolution of network parameters hence induces an evolving attractor landscape, a process which we term attractor metadynamics. We study the nature of the metadynamics of the attractor landscape for several continuous-time autonomous model networks. We find both first- and second-order changes in the location of adiabatic attractors and argue that the study of the continuously evolving attractor landscape constitutes a powerful tool for understanding the overall development of the neural dynamics

Phase Transitions of Neural Networks

The cooperative behaviour of interacting neurons and synapses is studied using models and methods from statistical physics. The competition between training error and entropy may lead to discontinuous properties of the neural network. This is demonstrated for a few examples: Perceptron, associative memory, learning from examples, generalization, multilayer networks, structure recognition, Bayesian estimate, on-line training, noise estimation and time series generation.Comment: Plenary talk for MINERVA workshop on mesoscopics, fractals and neural networks, Eilat, March 1997 Postscript Fil

Computational physics of the mind

In the XIX century and earlier such physicists as Newton, Mayer, Hooke, Helmholtz and Mach were actively engaged in the research on psychophysics, trying to relate psychological sensations to intensities of physical stimuli. Computational physics allows to simulate complex neural processes giving a chance to answer not only the original psychophysical questions but also to create models of mind. In this paper several approaches relevant to modeling of mind are outlined. Since direct modeling of the brain functions is rather limited due to the complexity of such models a number of approximations is introduced. The path from the brain, or computational neurosciences, to the mind, or cognitive sciences, is sketched, with emphasis on higher cognitive functions such as memory and consciousness. No fundamental problems in understanding of the mind seem to arise. From computational point of view realistic models require massively parallel architectures