Reduction of a model of an excitable cell to a one-dimensional map

Medvedev, Georgi S.

doi:10.1016/j.physd.2005.01.021

Cited by 39 publications

(61 citation statements)

References 33 publications

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“…It is worth noting that one-dimensional maps are increasingly used to model more complex neuronal dynamics like those of the Hodgkin-Huxley neuron model [57]. Here we propose and give evidence that parametrically coupled maps can be used to model the dynamics of high-dimensional nonlinear dynamical systems like the cortico-thalamocortical loop and we introduce the concept of cortex-module that paves the way to creating controlled trajectories in state space that may facilitate perceptual categorization.…”

Section: Finite Reproduction Time Refractory Periodmentioning

confidence: 95%

Corticonic models of brain mechanisms underlying cognition and intelligence

Farhat

2007

Physics of Life Reviews

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The concern of this review is brain theory or more specifically, in its first part, a model of the cerebral cortex and the way it:(a) interacts with subcortical regions like the thalamus and the hippocampus to provide higherlevel-brain functions that underlie cognition and intelligence, (b) handles and represents dynamical sensory patterns imposed by a constantly changing environment, (c) copes with the enormous number of such patterns encountered in a lifetime bymeans of dynamic memory that offers an immense number of stimulusspecific attractors for input patterns (stimuli) to select from, (d) selects an attractor through a process of "conjugation" of the input pattern with the dynamics of the thalamo-cortical loop, (e) distinguishes between redundant (structured)and non-redundant (random) inputs that are void of information, (f) can do categorical perception when there is access to vast associative memory laid out in the association cortex with the help of the hippocampus, and (g) makes use of "computation" at the edge of chaos and information driven annealing to achieve all this. Other features and implications of the concepts presented for the design of computational algorithms and machines with brain-like intelligence are also discussed. The material and results presented suggest, that a Parametrically Coupled Logistic Map network (PCLMN) is a minimal model of the thalamo-cortical complex and that marrying such a network to a suitable associative memory with reentry or feedback forms a useful, albeit, abstract model of a cortical module of the brain that could facilitate building a simple artificial brain. In the second part of the review, the results of numerical simulations and drawn conclusions in the first part are linked to the most directly relevant works and views of other workers. What emerges is a picture of brain dynamics on the mesoscopic and macroscopic scales that gives a glimpse of the nature of the long sought after brain code underlying intelligence and other higher level brain functions. AbstractThe concern of this review is brain theory or more specifically, in its first part, a model of the cerebral cortex and the way it: (a) interacts with subcortical regions like the thalamus and the hippocampus to provide higher-level-brain functions that underlie cognition and intelligence, (b) handles and represents dynamical sensory patterns imposed by a constantly changing environment, (c) copes with the enormous number of such patterns encountered in a lifetime by means of dynamic memory that offers an immense number of stimulus-specific attractors for input patterns (stimuli) to select from, (d) selects an attractor through a process of "conjugation" of the input pattern with the dynamics of the thalamo-cortical loop, (e) distinguishes between redundant (structured) and non-redundant (random) inputs that are void of information, (f) can do categorical perception when there is access to vast associative memory laid out in the association cortex with the help of the hippocampus, and (g) ma...

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Section: Finite Reproduction Time Refractory Periodmentioning

confidence: 95%

Corticonic models of brain mechanisms underlying cognition and intelligence

Farhat

2007

Physics of Life Reviews

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“…A number of different strategies for performing such reductions have been proposed [20,35,36], all of which could be equivalently employed. Our analysis relied on straightforward Poincaré mapping and fast-slow subsystem decomposition [25,35].…”

Section: B Unidimensional Reductions Of Neuronal Modelmentioning

confidence: 99%

“…Our analysis relied on straightforward Poincaré mapping and fast-slow subsystem decomposition [25,35].…”

Section: B Unidimensional Reductions Of Neuronal Modelmentioning

confidence: 99%

Noise, transient dynamics, and the generation of realistic interspike interval variation in square-wave burster neurons

et al. 2014

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First return maps of interspike intervals for biological neurons that generate repetitive bursts of impulses can display stereotyped structures (neuronal signatures). Such structures have been linked to the possibility of multicoding and multifunctionality in neural networks that produce and control rhythmical motor patterns. In some cases, isolating the neurons from their synaptic network reveals irregular, complex signatures that have been regarded as evidence of intrinsic, chaotic behavior.We show that incorporation of dynamical noise into minimal neuron models of square-wave bursting (either conductance-based or abstract) produces signatures akin to those observed in biological examples, without the need for fine-tuning of parameters or ad hoc constructions for inducing chaotic activity. The form of the stochastic term is not strongly constrained, and can approximate several possible sources of noise, e.g. random channel gating or synaptic bombardment.The cornerstone of this signature generation mechanism is the rich, transient, but deterministic dynamics inherent in the square-wave (saddle-node/homoclinic) mode of neuronal bursting. We show that noise causes the dynamics to populate a complex transient scaffolding or skeleton in state space, even for models that (without added noise) generate only periodic activity (whether in bursting or tonic spiking mode).

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“…In this section, we derive a one-dimensional map representing the dynamics of a self-coupled cell given by (1-4), directly following Medvedev (2005). We subsequently analyze its properties and consider the dynamics of the map, with particular attention to transitions between bursting and spiking.…”

Section: Self-coupled Cell: One-dimensional Mapmentioning

confidence: 99%

“…Maps have also been constructed phenomenologically to reproduce some important characteristics of a given activity pattern (Wang 1991;Rulkov 2002Rulkov , 2004. In this work, we follow a rigorous reduction, derived recently by Medvedev (2005) for individual square-wave burst-capable elements such as the single-cell Butera et al (1999a) preBötC model, that takes advantage of the presence of two disparate timescales in the model. We recapitulate Medvedev's approach to show explicitly how it plays out for a self-coupled model preBötc cell.…”

Section: Introductionmentioning

confidence: 99%

First return maps for the dynamics of synaptically coupled conditional bursters

2010

Self Cite

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The pre-Bötzinger complex (preBötc) in the mammalian brainstem has an important role in generating respiratory rhythms. An influential differential equation model for the activity of individual neurons in the preBötc yields transitions from quiescence to bursting to tonic spiking as a parameter is varied. Further, past work has established that bursting dynamics can arise from a pair of tonic model cells coupled with synaptic excitation. In this paper, we analytically derive one- and two-dimensional maps from the differential equations for a self-coupled neuron and a two-neuron network, respectively. Using a combination of analysis and simulations of these maps, we explore the possible forms of dynamics that the model networks can produce as well as which transitions between dynamic regimes are mathematically possible.

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Reduction of a model of an excitable cell to a one-dimensional map

Cited by 39 publications

References 33 publications

Corticonic models of brain mechanisms underlying cognition and intelligence

Corticonic models of brain mechanisms underlying cognition and intelligence

Noise, transient dynamics, and the generation of realistic interspike interval variation in square-wave burster neurons

First return maps for the dynamics of synaptically coupled conditional bursters

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