Weakly Connected Neural Networks

Hoppensteadt, Frank C.; Izhikevich, Eugene M.

doi:10.1007/978-1-4612-1828-9

Cited by 996 publications

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“…To interact they should establish certain resonant relation between their frequencies. This fact was proven by Hoppensteadt and Izhikevich (1997) for the case of periodic rhythmic activity and later generalized by Izhikevich (1998) for the case of quasiperiodic (multi-frequency) rhythmic activity. We do not present the proofs here, since they involve normally hyperbolic invariant manifold, averaging, and KAM (Kolmogorov -ArnoldMoser) theories.…”

Section: The Nature Of Neural Codementioning

confidence: 81%

“…In contrast, our goal is to derive results that are largely independent of the model and that can be observed in a broad class or a family of neural models. To carry out this task we reduce the entire family of models to a canonical model (Hoppensteadt and Izhikevich 1997). Briefly, a model is canonical for the family if every model from the family can be transformed into this one by a continuous change of variables.…”

Section: The Canonical Model Approachmentioning

confidence: 99%

“…Some partial success is achieved when the family describes neurons, columns or brain structures operating in a critical regime, such as near a bifurcation (Hoppensteadt and Izhikevich 1997). Then, the question of biological plausibility of a canonical model is replaced by the question of plausibility of a critical regime.…”

Section: The Canonical Model Approachmentioning

confidence: 99%

“…This assumption is discussed in detail by Hoppensteadt and Izhikevich (1997) and Izhikevich (1998). It is based on the observation that average amplitude of post synaptic potential is smaller than 1 mV, which implies that there must be many (a few hundred) presynaptic neurons firing simultaneously to make a given cell fire (McNaughton et al, 1981).…”

Section: Weak Connectionsmentioning

confidence: 99%

“…Malkin, 1949Malkin, , 1956Blechman, 1971;Neu, 1979;Ermentrout, 1981;Kuramoto, 1984;Ermentrout and Kopell, 1991) that the entire system (1, 2) can be approximated by a phase model that is defined on the n+ 1 dimensional torus T n + 1 . Hoppensteadt and Izhikevich (1997) have gone further and proved that there is actually a continuous change of variables that transforms (1, 2) into the phase model. Below is the precise statement of this result.…”

Section: Autonomous Oscillationsmentioning

confidence: 99%

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

1998

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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.

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Section: The Nature Of Neural Codementioning

confidence: 81%

Section: The Canonical Model Approachmentioning

confidence: 99%

Section: The Canonical Model Approachmentioning

confidence: 99%