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We discuss the biological plausibility and computational efficiency of some of the most useful models of spiking and bursting neurons. We compare their applicability to large-scale simulations of cortical neural networks. Index Terms--Chaos, Hodgkin-Huxley, pulse-coupled neural network (PCNN), quadratic integrate-and-fire (I&F), spike-timing.

A model is presented that reproduces spiking and bursting behavior of known types of cortical neurons. The model combines the biologically plausibility of Hodgkin-Huxley-type dynamics and the computational efficiency of integrate-and-fire neurons. Using this model, one can simulate tens of thousands of spiking cortical neurons in real time (1 ms resolution) using a desktop PC. Index Terms--Bursting, cortex, Hodgkin-Huxley, PCNN, quadratic integrate-and-fire, spiking, thalamus.

What is the functional significance of generating a burst of spikes, as opposed to a single spike? A dominant point of view is that bursts are needed to increase the reliability of communication between neurons. Here, we discuss the alternative, but complementary, hypothesis: bursts with specific resonant interspike frequencies are more likely to cause a postsynaptic cell to fire than are bursts with higher or lower frequencies. Such a frequency preference might occur at the level of individual synapses because of the interplay between short-term synaptic depression and facilitation, or at the postsynaptic cell level because of subthreshold membrane potential oscillations and resonance. As a result, the same burst could resonate for some synapses or cells and not resonate for others, depending on their natural resonance frequencies. This observation suggests that, in addition to increasing reliability of synaptic transmission, bursts of action potentials might provide effective mechanisms for selective communication between neurons.

We combine here two well-known and established concepts: microelectromechanical systems (MEMS) and neurocomputing. First, we consider MEMS oscillators having low amplitude activity and we derive a simple mathematical model that describes nonlinear phase-locking dynamics in them. Then, we investigate a theoretical possibility of using MEMS oscillators to build an oscillatory neurocomputer having autocorrelative associative memory. The neurocomputer stores and retrieves complex oscillatory patterns in the form of synchronized states with appropriate phase relations between the oscillators. Thus, we show that MEMS alone can be used to build a sophisticated information processing system (U.S. provisional patent 60/178,654). Index Terms--Andronov--Hopf bifurcation, [Mu] resonantors, neural networks, oscillatory associative memory, smart matter.

We propose a novel architecture of an oscillatory neural network that consists of phase-locked loop (PLL) circuits. It stores and retrieves complex oscillatory patterns as synchronized states with appropriate phase relations between neurons. Index Terms--Brain rhythms, oscillatory associative memory, temporal pattern recognition, voltage-controlled oscillators (VCO's).

We study pulse-coupled neural networks that satisfy only two assumptions: each isolated neuron fires periodically, and the neurons are weakly connected. Each such network can be transformed by a piece-wise continuous change of variables into a phase model, whose synchronization behavior and oscillatory associative properties are easier to analyze and understand. Using the phase model, we can predict whether a given pulse-coupled network has oscillatory associative memory, or what minimal adjustments should be made so that it can acquire memory. In the search for such minimal adjustments we obtain a large class of simple pulse-coupled neural networks that can memorize and reproduce synchronized temporal patterns the same way a Hopfield network does with static patterns. The learning occurs via modification of synaptic weights and/or synaptic transmission delays. Index Terms--Canonical models, Class 1 neural excitability, integrate-and-fire neurons, multiplexing, syn-fire chain, transmission delay.

Many scientists believe that all pulse-coupled neural networks are toy models that are far away from the biological reality. We show here, however, that a huge class of biophysically detailed and biologically plausible neural-network models can be transformed into a canonical pulse-coupled form by a piece-wise continuous, possibly noninvertible, change of variables. Such transformations exist when a network satisfies a number of conditions; e.g., it is weakly connected; the neurons are Class 1 excitable (i.e., they can generate action potentials with an arbitrary small frequency); and the synapses between neurons are conventional (i.e., axo-dendritic and axo-somatic). Thus, the difference between studying the pulse-coupled model and Hodgkin-Huxley-type neural networks is just a matter of a coordinate change. Therefore, any piece of information about the pulse-coupled model is valuable since it tells something about all weakly connected networks of Class 1 neurons. For example, we show that the pulse-coupled network of identical neurons does not synchronize in-phase. This confirms Ermentrout's result that weakly connected Class 1 neurons are difficult to synchronize, regardless of the equations that describe dynamics of each cell. Index Terms--Canonical model, class 1 neural excitability, conventional synapses, desynchronization, integrate-and-fire, saddle-node on limit cycle bifurcation, weakly connected neural networks.