Circuits that contain the Model Concept : Bifurcation

(A bifurcation is a sudden qualitative change in the behavior of a dynamical system caused by a change in a parameter(s).)
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    Models   Description
1. A Moth MGC Model-A HH network with quantitative rate reduction (Buckley & Nowotny 2011)
We provide the model used in Buckley & Nowotny (2011). It consists of a network of Hodgkin Huxley neurons coupled by slow GABA_B synapses which is run alongside a quantitative reduction described in the associated paper.
2. Bursting and oscillations in RD1 Retina driven by AII Amacrine Neuron (Choi et al. 2014)
"In many forms of retinal degeneration, photoreceptors die but inner retinal circuits remain intact. In the rd1 mouse, an established model for blinding retinal diseases, spontaneous activity in the coupled network of AII amacrine and ON cone bipolar cells leads to rhythmic bursting of ganglion cells. Since such activity could impair retinal and/or cortical responses to restored photoreceptor function, understanding its nature is important for developing treatments of retinal pathologies. Here we analyzed a compartmental model of the wild-type mouse AII amacrine cell to predict that the cell's intrinsic membrane properties, specifically, interacting fast Na and slow, M-type K conductances, would allow its membrane potential to oscillate when light-evoked excitatory synaptic inputs were withdrawn following photoreceptor degeneration. ..."
3. Irregular spiking in NMDA-driven prefrontal cortex neurons (Durstewitz and Gabriel 2006)
Slow N-Methyl-D-aspartic acid (NMDA) synaptic currents are assumed to strongly contribute to the persistently elevated firing rates observed in prefrontal cortex (PFC) during working memory. During persistent activity, spiking of many neurons is highly irregular. ... The highest interspike-interval (ISI) variability occurred in a transition regime where the subthreshold membrane potential distribution shifts from mono- to bimodality, ... Predictability within irregular ISI series was significantly higher than expected from a noise-driven linear process, indicating that it might best be described through complex (potentially chaotic) nonlinear deterministic processes. Accordingly, the phenomena observed in vitro could be reproduced in purely deterministic biophysical model neurons. High spiking irregularity in these models emerged within a chaotic, close-to-bifurcation regime characterized by a shift of the membrane potential distribution from mono- to bimodality and by similar ISI return maps as observed in vitro. ... NMDA-induced irregular dynamics may have important implications for computational processes during working memory and neural coding.
4. Leech Heart Interneuron model (Sharma et al 2020)
Fractional order Leech heart interneuron model is investigated. Different firing properties are explored. In this article, we investigate the alternation of spiking and bursting phenomena of an uncoupled and coupled fractional Leech-Heart (L-H) neurons. We show that a complete graph of heterogeneous de-synchronized neurons in the backdrop of diverse memory settings (a mixture of integer and fractional exponents) can eventually lead to bursting with the formation of cluster synchronization over a certain threshold of coupling strength, however, the uncoupled L-H neurons cannot reveal bursting dynamics. Using the stability analysis in fractional domain, we demarcate the parameter space where the quiescent or steady-state emerges in uncoupled L-H neuron. Finally, a reduced-order model is introduced to capture the activities of the large network of fractional-order model neurons.
5. Mean-field models of neural populations under electrical stimulation (Cakan & Obermayer 2020)
Weak electrical inputs to the brain in vivo using transcranial electrical stimulation or in isolated cortex in vitro can affect the dynamics of the underlying neural populations. However, it is poorly understood what the exact mechanisms are that modulate the activity of neural populations as a whole and why the responses are so diverse in stimulation experiments. Despite this, electrical stimulation techniques are being developed for the treatment of neurological diseases in humans. To better understand these interactions, it is often necessary to simulate and analyze very large networks of neurons, which can be computationally demanding. In this theoretical paper, we present a reduced model of coupled neural populations that represents a piece of cortical tissue. This efficient model retains the dynamical properties of the large network of neurons it is based on while being several orders of magnitude faster to simulate. Due to the biophysical properties of the neuron model, an electric field can be coupled to the population. We show that weak electric fields often used in stimulation experiments can lead to entrainment of neural oscillations on the population level, and argue that the responses critically depend on the dynamical state of the neural system.
6. Minimal model of interictal and ictal discharges “Epileptor-2” (Chizhov et al 2018)
"Seizures occur in a recurrent manner with intermittent states of interictal and ictal discharges (IIDs and IDs). The transitions to and from IDs are determined by a set of processes, including synaptic interaction and ionic dynamics. Although mathematical models of separate types of epileptic discharges have been developed, modeling the transitions between states remains a challenge. A simple generic mathematical model of seizure dynamics (Epileptor) has recently been proposed by Jirsa et al. (2014); however, it is formulated in terms of abstract variables. In this paper, a minimal population-type model of IIDs and IDs is proposed that is as simple to use as the Epileptor, but the suggested model attributes physical meaning to the variables. The model is expressed in ordinary differential equations for extracellular potassium and intracellular sodium concentrations, membrane potential, and short-term synaptic depression variables. A quadratic integrate-and-fire model driven by the population input current is used to reproduce spike trains in a representative neuron. ..."

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