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F C Hoppensteadt

Publications and source records attributed to F C Hoppensteadt.

15 recordsLinked to original sources
PubMed2001

Phase clustering and transition to phase synchronization in a large number of coupled nonlinear oscillators.

The transition to phase synchronization in systems consisting of a large number (N) of coupled nonlinear oscillators via the route of phase clustering (phase synchronization among subsets of oscillators) is investigated. We elucidate the mechanism for the merger of phase clusters and find an algebraic scaling between the critical coupling parameter required for phase synchronization and N. Our result implies that, in realistic situations, phase clustering may be more prevalent than full phase synchronization.

Journal Article↗
PubMed2001

Mathematical models and simulations of bacterial growth and chemotaxis in a diffusion gradient chamber.

The diffusion gradient chamber (DGC) is a novel device developed to study the response of chemotactic bacteria to combinations of nutrients and attractants [7]. Its purpose is to characterize genetic variants that occur in many biological experiments. In this paper, a mathematical model which describes the spatial distribution of a bacterial population within the DGC is developed. Mathematical analysis of the model concerning positivity and boundedness of the solutions are given. An ADI (Alternating Direction Implicit) method is constructed for finding numerical solutions of the model and carrying out computer simulations. The numerical results of the model successfully reproduced the patterns that were observed in the experiments using the DGC.

Bacteria↗
PubMed1996

Synaptic organizations and dynamical properties of weakly connected neural oscillators. I. Analysis of a canonical model.

We study weakly connected networks of neural oscillators near multiple Andronov-Hopf bifurcation points. We analyze relationships between synaptic organizations (anatomy) of the networks and their dynamical properties (function). Our principal assumptions are: (1) Each neural oscillator comprises two populations of neurons; excitatory and inhibitory ones; (2) activity of each population of neurons is described by a scalar (one-dimensional) variable; (3) each neural oscillator is near a nondegenerate supercritical Andronov-Hopf bifurcation point; (4) the synaptic connections between the neural oscillators are weak. All neural networks satisfying these hypotheses are governed by the same dynamical system, which we call the canonical model. Studying the canonical model shows that: (1) A neural oscillator can communicate only with those oscillators which have roughly the same natural frequency. That is, synaptic connections between a pair of oscillators having different natural frequencies are functionally insignificant. (2) Two neural oscillators having the same natural frequencies might not communicate if the connections between them are from among a class of pathological synaptic configurations. In both cases the anatomical presence of synaptic connections between neural oscillators does not necessarily guarantee that the connections are functionally significant. (3) There can be substantial phase differences (time delays) between the neural oscillators, which result from the synaptic organization of the network, not from the transmission delays. Using the canonical model we can illustrate self-ignition and autonomous quiescence (oscillator death) phenomena. That is, a network of passive elements can exhibit active properties and vice versa. We also study how Dale's principle affects dynamics of the networks, in particular, the phase differences that the network can reproduce. We present a complete classification of all possible synaptic organizations from this point of view. The theory developed here casts some light on relations between synaptic organization and functional properties of oscillatory networks. The major advantage of our approach is that we obtain results about all networks of neural oscillators, including the real brain. The major drawback is that our findings are valid only when the brain operates near a critical regime, viz. for a multiple Andronov-Hopf bifurcation.

Cell Death↗
PubMed1996

Synaptic organizations and dynamical properties of weakly connected neural oscillators. II. Learning phase information.

This is the second of two articles devoted to analyzing the relationship between synaptic organizations (anatomy) and dynamical properties (function) of networks of neural oscillators near multiple supercritical Andronov-Hopf bifurcation points. Here we analyze learning processes in such networks. Regarding learning dynamics, we assume (1) learning is local (i.e. synaptic modification depends on pre- and postsynaptic neurons but not on others), (2) synapses modify slowly relative to characteristic neuron response times, (3) in the absence of either pre- or postsynaptic activity, the synapse weakens (forgets). Our major goal is to analyze all synaptic organizations of oscillatory neural networks that can memorize and retrieve phase information or time delays. We show that such network have the following attributes: (1) the rate of synaptic plasticity connected with learning is determined locally by the presynaptic neurons, (2) the excitatory neurons must be long-axon relay neurons capable of forming distant connections with other excitatory and inhibitory neurons, (3) if inhibitory neurons have long axons, then the network can learn, passively forget and actively unlearn information by adjusting synaptic plasticity rates.

Learning↗
PubMed1989

Intermittent chaos, self-organization, and learning from synchronous synaptic activity in model neuron networks.

Self-organization of frequencies is studied by using model neurons called VCONs (voltage-controlled oscillator neuron models). These models give direct access to frequency information, in contrast to all-or-none neuron models, and they generate voltage spikes that phase-lock to oscillatory stimulation, similar to phase-locking of action potentials to oscillatory voltage stimulation observed in Hodgkin-Huxley preparations of squid axons. The rotation vector method is described and used to study how networks synchronize, even in the presence of noise or when damaged; the entropy of ratios of phases is used to construct an energy function that characterizes organized behavior. Computer simulations show that rotation numbers (output frequency/input frequency) describe both chaotic and nonchaotic behavior. Learning occurs when synaptic connections strengthen in response to stimulation that is synchronous with cell activity. It is shown that intermittent chaotic firing is suppressed and simple stable responses are enhanced by such learning in VCON networks. This analysis provides a rigorous basis for further investigation of the ideas of Wiener [Wiener, N. (1961) Cybernetics (MIT Press, Cambridge, MA), p. 191] on the origin of slow brain waves due to "the pulling together of frequencies."

Action Potentials↗
PubMed1987

A mathematical analysis of small mammal populations.

Populations of Microtus montanus, the montane vole, have been extensively studied. It is known that their reproductive activity is closely linked to the availability of the chemicals in growing plants. We use a mathematical model here to study how the length of the vegetative season and the natural reproduction rhythm of voles are involved in the long term dynamics of the population numbers. In particular, we use data obtained from Timpie Springs, Utah, and from Jackson Hole, Wyoming, to formulate a model. The novelty of this model is its use of littering curves that highlight the temporally discrete nature of vole reproduction. The model shows how the timing of the vegetative season can influence vole population sizes.

Animals↗
PubMed1987

Frequency modulation dynamics in neural networks.

The VCON model described here shares certain qualitative features of stimulus-response characteristics with the data in FIGURE 1 and TABLE 1. At the same time, it provides an uncomplicated methodology for modelling neural networks that emphasizes their frequency aspects. In particular, models based on VCONs are amenable to the rotation vector method, which can be used to uncover stable synchronization of firing within a network. This is described in the appendix, where it is shown that the stable firing patterns within the network correspond to minima of an associated (local) energy function. We have seen here how a model of a simple CPG for breathing can be constructed and analyzed. Similar models for rhythm splitting of small mammal activity cycles, sound location networks, and motility in the gastrointestinal tract have been constructed, and large networks of VCONs have been shown to have stable spatial patterns of synchronization.

Action Potentials↗
PubMed1984

Synchronized oscillations in networks of neuron analogue circuits.

A simplified circuit analogue of a nerve cell is introduced here. This circuit is based on modulation of a voltage-controlled oscillator by signals entering through a circuit analogue of a chemical synapse. Phase locking of frequency-encoded information is described, and it is shown that large networks can sustain stable spatial patterns of phase-locked behaviour.

Animals↗
PubMed1982

Phase locking of biological clocks.

Radial isochron clocks (RICs) and their response to external signals and coupling with other RICs are studied. RICs are derived as phase approximations to self-sustained oscillators. Their response to single impulses (phase resetting) and to repetitive impulses is determined. This response may be harmonic or chaotic. Finally, the effect of coupling between clocks is studied. Simple coupling is shown to exhibit rhythm splitting like that observed in fish and small mammals. New phase locking results for general weakly coupled RIC systems are also derived.

Animals↗
PubMed1976

Synchronization of periodical cicada emergences.

Synchronized insect emergences are shown to be a possible consequence of predation in the presence of a limited environmental carrying capacity through a mathematical model for cicada populations that includes these two features. Synchronized emergences, like those observed in 13- and 17-year cicades, are predicted for insects with sufficiently long life-spans. Balanced solutions, in which comparable emergences occur each year, are found for insects having sufficiently short life-spans, such as 3-, 4-, and 7-year cicadas. For the values used here, synchronized emergences occur for insects with life-spans of 10 years or more, and balanced emergences occur for life-spans of fewer than 10 years.

Animals↗
PubMedDate not supplied

Memorizing and recalling spatial-temporal patterns in an oscillator model of the hippocampus.

We describe the model of the hippocampus consisting of interactive oscillators with input from the entorhinal cortex (modulating the main information flow by a theta rhythm) and the septum (a theta rhythm generator). When interconnections between oscillators are allowed to strengthen in an adaptive way, the network can be trained using a series of lessons. This results in a connection matrix that memorizes the temporal sequence of inputs. Presenting one of the lessons to the trained network results in reproduction of the remainder of the sequence. In this paper, we create such a connection matrix, derive from it an appropriate Markov chain and simulate the chain to illustrate its dynamics.

Hippocampus↗
PubMedDate not supplied

Thalamo-cortical interactions modeled by weakly connected oscillators: could the brain use FM radio principles?

We consider all models of the thalamo-cortical system that satisfy the following two assumptions: (1) each cortical column is an autonomous oscillator; (2) connections between cortical columns and the thalamus are weak. Our goal is to deduce from these assumptions general principles of thalamo-cortical interactions that are independent of the equations describing the system. We find that the existence of synaptic connections between any two cortical columns does not guarantee that the columns interact: They interact only when there is a certain nearly resonant relation between their frequencies, which implies that the interactions are frequency modulated (FM). When the resonance relation holds, the cortical columns interact through phase modulations. Thus, communications between weakly connected cortical oscillators employ a principle similar to that in FM radio: The frequency of oscillation encodes the channel of communication, while the information is transmitted via phase modulations. If the thalamic input has an appropriate frequency, then it can dynamically link any two cortical columns, even those that have non-resonant frequencies and would otherwise be unlinked. Thus, by adjusting its temporal activity, the thalamus has control over information processing taking place in the cortex. Our results suggest that the mean firing rate (frequency) of periodically spiking neuron does not carry any information other than identifying a channel of communication. Information (i.e. neural code) is carried through modulations of interspike intervals.

Cerebral Cortex↗