arXiv subjects
Publications and source records attributed to John Hertz.
We present a dynamical description and analysis of non-equilibrium transitions in the noisy Ginzburg-Landau equation based on a canonical phase space formulation. The transition pathways are characterized by nucleation and subsequent propagation of domain walls or solitons. We also evaluate the Arrhenius factor in terms of an associated action and find good agreement with recent numerical optimization studies.
We use mean field theory to study the response properties of a simple randomly-connected model cortical network of leaky integrate-and-fire neurons with balanced excitation and inhibition. The formulation permits arbitrary temporal variation of the input to the network and takes exact account of temporal firing correlations. We find that neuronal firing statistics depend sensitively on the firing threshold. In particular, spike count variances can be either significantly greater than or significantly less than those for a Poisson process. These findings may help in understanding the variability observed experimentally in cortical neuronal responses.
We introduce a model of generalized Hebbian learning and retrieval in oscillatory neural networks modeling cortical areas such as hippocampus and olfactory cortex. Recent experiments have shown that synaptic plasticity depends on spike timing, especially on synapses from excitatory pyramidal cells, in hippocampus and in sensory and cerebellar cortex. Here we study how such plasticity can be used to form memories and input representations when the neural dynamics are oscillatory, as is common in the brain (particularly in the hippocampus and olfactory cortex). Learning is assumed to occur in a phase of neural plasticity, in which the network is clamped to external teaching signals. By suitable manipulation of the nonlinearity of the neurons or of the oscillation frequencies during learning, the model can be made, in a retrieval phase, either to categorize new inputs or to map them, in a continuous fashion, onto the space spanned by the imprinted patterns. We identify the first of these possibilities with the function of olfactory cortex and the second with the observed response characteristics of place cells in hippocampus. We investigate both kinds of networks analytically and by computer simulations, and we link the models with experimental findings, exploring, in particular, how the spike timing dependence of the synaptic plasticity constrains the computational function of the network and vice versa.
We study the Langevin dynamics of the standard random heteropolymer model by mapping the problem to a supersymmetric field theory using the Martin-Siggia-Rose formalism. The resulting model is solved non-perturbatively employing a Gaussian variational approach. In constructing the solution, we assume that the chain is very long and impose the translational invariance which is expected to be present in the bulk of the globule by averaging over the center the of mass coordinate. In this way we derive equations of motion for the correlation and response functions C(t,t') and R(t,t'). The order parameters are extracted from the asymptotic behavior of these functions. We find a dynamical phase diagram with frozen (glassy) and melted (ergodic) phases. In the glassy phase the system fails to reach equilibrium and exhibits aging of the type found in p-spin glasses. Within the approximations used in this study, the random heteropolymer model can be mapped to the problem of a manifold in a random potential with power law correlations.
We present a model of an olfactory system that performs odor segmentation. Based on the anatomy and physiology of natural olfactory systems, it consists of a pair of coupled modules, bulb and cortex. The bulb encodes the odor inputs as oscillating patterns. The cortex functions as an associative memory: When the input from the bulb matches a pattern stored in the connections between its units, the cortical units resonate in an oscillatory pattern characteristic of that odor. Further circuitry transforms this oscillatory signal to a slowly-varying feedback to the bulb. This feedback implements olfactory segmentation by suppressing the bulbar response to the pre-existing odor, thereby allowing subsequent odors to be singled out for recognition.
We present a model of a coupled system of the olfactory bulb and cortex. Odor inputs to the epithelium are transformed to oscillatory bulbar activities. The cortex recognizes the odor by resonating to the bulbar oscillating pattern when the amplitude and phase patterns from the bulb match an odor memory stored in the intracortical synapses. We assume a cortical structure which transforms the odor information in the oscillatory pattern to a slow DC feedback signal to the bulb. This feedback suppresses the bulbar response to the pre-existing odor, allowing subsequent odor objects to be segmented out for recognition.
We study the off-equilibrium relaxational dynamics of the Amit-Roginsky $ϕ^3$ field theory, for which the mode coupling approximation is exact. We show that complex phenomena such as aging and ergodicity breaking are present at low temperature, similarly to what is found in long range spin glasses. This is a generalization of mode coupling theory of the structural glass transition to off-equilibrium situations.