Statistical model of the hippocampal CA3 region. I. The single-cell module: bursting model of the pyramidal cell.

A model with intermediate complexity is introduced to reproduce the basic firing modes of the CA3 pyramidal cell. Our model consists of a single compartment, has two variables (membrane potential and internal calcium concentration), and involves two separate stages for interspike mechanisms and firing. Interspike dynamics is governed by voltage- and calcium-dependent ionic channels but no channel kinetics are provided. This model is suitable to be included in our statistical population model (Part II, following paper). Bifurcation analysis reveals that interspike dynamics rather than sodium firing has the dominant role in the control of bursting/nonbursting behavior.

Statistical model of the hippocampal CA3 region II. The population framework: model of rhythmic activity in the CA3 slice.

A statistical model is given to describe the electrical activity patterns of large neural populations of the hippocampal CA3 region. A continuous model has been formalized to describe the statistical processes governing the interactions within and between neural fields. The system of partial differential equations contains diffusion terms which determine the evolution of second moments of the probability distribution functions. The model is supplemented with a differential description of post-synaptic potentials. The discretization procedure has been designed so as to make the discrete equations scaling invariant. Population activities as well as underlying single-cell voltages are simulated during normal and epileptiform activities in the hippocampal CA3 slice. It is demonstrated that our model can reproduce electrophysiological phenomena characteristic to both single-cell and population activities. Specifically, fully synchronized population bursts, synchronized synaptic potentials, and low amplitude population oscillation were obtained.

Rhythmogenesis in single cells and population models: olfactory bulb and hippocampus.

Dynamics of single cells, small networks and large cell populations are the subject of investigation. The generation and propagation of action potentials in the two major cell types of the olfactory bulb, i.e. in the mitral and granule cells, are simulated by multi-compartmental modeling techniques. The specific effects of the individual ionic currents, the propagation of the signals through the compartments, and dynamic phenomena occurring in small networks (such as synchronized oscillation due to excitatory and inhibitory coupling) have been demonstrated. A statistical model is given to describe the electrical activity patterns of large neural populations. The model is applied for describing the CA3 region of the hippocampus by incorporating some basic electrophysiological properties of hippocampal pyramidal and inhibitory neurons. Population activities as well as underlying single cell voltages are simulated during population bursts in the model of disinhibited hippocampal CA3 slice.

Dynamics of the olfactory bulb: bifurcations, learning, and memory.

A mathematical model for describing dynamic phenomena in the olfactory bulb is presented. The nature of attractors and the bifurcation sequences in terms of the lateral connection strength in the mitral layer are studied numerically. Chaotic activity has only been found in the case of strong excitatory coupling. Synaptic modification-induced transition from oscillation to chaos is demonstrated. A model for a simple associative memory is also presented.

On the dynamic organization of memory. A mathematical model of associative free recall.

The interaction of memory structures and retrieval dynamics is discussed. A mathematical model for associative free recall is presented to support the view that the organization of simple processing units plays an important role in the retrieval of memory traces. Computer simulations show that "flexibility" and "fidelity" of the dynamics strongly depend on the network structure, the amplification and decay parameters, and the noise term.

Dynamic phenomena in the olfactory bulb.

A mathematical model of the olfactory bulb is presented to study the dynamics of the bulbar information processing. A two level model is adopted to describe both neural activity and synaptic modifiability. The model takes explicitly into account the existence of lateral interactions in the mitral layer, and the synaptic modifiability of these connections. A series of bifurcation phenomena among fix points, limit cycle and strange attractors have been demonstrated. Chaos occurred only in the case of excitatory lateral connections. Coexistence between oscillation and chaos, and synaptic modification induced transition have also been found. The model attempts to demonstrate the associative memory character of the olfactory bulb. Odour qualities are coded in distributed spatial amplitude patterns. Differential equations for the mitral and granule cell activities have been supplemented by a continuous-time local learning rule. A nonlinear forgetting term and a selective decreasing term is added to the Hebbian learning rule. A learned odour can be recalled by a subset of the pattern. There is a strict restriction on the parameters: only those values can be admitted which generate physiologically justified activity signals.