Models that contain the Cell : Abstract Morris-Lecar neuron

(A dynamical system of two nonlinear equations that is a simplified model of the higher dimensional Hodgkin-Huxley neuron.)
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    Models   Description
1.  A model for how correlation depends on the neuronal excitability type (Hong et al. 2012)
“ … Using simulations and experiments in rat hippocampal neurons, we show here that pairs of neurons receiving correlated input also exhibit correlations arising from precise spike-time synchronization. Contrary to rate comodulation, spike-time synchronization is unaffected by firing rate, thus enabling synchrony- and rate-based coding to operate independently. The type of output correlation depends on whether intrinsic neuron properties promote integration or coincidence detection: “ideal” integrators (with spike generation sensitive to stimulus mean) exhibit rate comodulation, whereas ideal coincidence detectors (with spike generation sensitive to stimulus variance) exhibit precise spike-time synchronization. … Our results explain how different types of correlations arise based on how individual neurons generate spikes, and why spike-time synchronization and rate comodulation can encode different stimulus properties. Our results also highlight the importance of neuronal properties for population-level coding insofar as neural networks can employ different coding schemes depending on the dominant operating mode of their constituent neurons. “
2.  Criticality,degeneracy in injury-induced changes in primary afferent excitability (Ratte et al 2014)
"Neuropathic pain remains notoriously difficult to treat despite numerous drug targets. Here, we offer a novel explanation for this intractability. Computer simulations predicted that qualitative changes in primary afferent excitability linked to neuropathic pain arise through a switch in spike initiation dynamics when molecular pathologies reach a tipping point (criticality), and that this tipping point can be reached via several different molecular pathologies (degeneracy). ..."
3.  Dorsal root ganglion (primary somatosensory) neurons (Rho & Prescott 2012)
In this paper, we demonstrate how dorsal root ganglion (DRG) neuron excitability can become pathologically altered, as occurs in neuropathic pain. Specifically, we reproduce pathological changes in spiking pattern (from transient to repetitive spiking) and the development of membrane potential oscillations and bursting.
4.  Dynamics of Spike Initiation (Prescott et al. 2008)
"Transduction of graded synaptic input into trains of all-or-none action potentials (spikes) is a crucial step in neural coding. Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties. Despite widespread use of this classification scheme, a generalizable explanation of its biophysical basis has not been described. We recorded from spinal sensory neurons representing each class and reproduced their transduction properties in a minimal model. With phase plane and bifurcation analysis, each class of excitability was shown to derive from distinct spike initiating dynamics. Excitability could be converted between all three classes by varying single parameters; moreover, several parameters, when varied one at a time, had functionally equivalent effects on excitability. From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin’s classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents. ..."
5.  Epileptic seizure model with Morris-Lecar neurons (Beverlin and Netoff 2011)
Here we use phase-response curves (PRC) from Morris-Lecar (M-L) model neurons with synaptic depression and gradually decrease input current to cells within a network simulation. This method effectively decreases firing rates resulting in a shift to greater network synchrony illustrating a possible mechanism of the transition phenomenon. PRCs are measured from the M-L conductance based model cell with a range of input currents within the limit cycle. A large network of 3000 excitatory neurons is simulated with a network topology generated from second-order statistics which allows a range of population synchrony. The population synchrony of the oscillating cells is measured with the Kuramoto order parameter, which reveals a transition from tonic to clonic phase exhibited by our model network.
6.  Frog second-order vestibular neuron models (Rossert et al. 2011)
This implements spiking Hodgkin-Huxley type models of tonic and phasic second-order vestibular neurons. Models fitted to intracellular spike and membrane potential recordings from frog (Rana temporaria). The models can be stimulated by intracellular step current, frequency current (ZAP) or synaptic stimulation.
7.  Homeostatic mechanisms may shape oscillatory modulations (Peterson & Voytek 2020)
"Neural oscillations are observed ubiquitously in the mammalian brain, but their stability is known to be rather variable. Some oscillations are tonic and last for seconds or even minutes. Other oscillations appear as unstable bursts. Likewise, some oscillations rely on excitatory AMPAergic synapses, but others are GABAergic and inhibitory. Why this diversity exists is not clear. We hypothesized Ca2+-dependent homeostasis could be important in finding an explanation. We tested this hypothesis in a highly simplified model of hippocampal neurons. In this model homeostasis profoundly alters the modulatory effect of neural oscillations. Under homeostasis, tonic AMPAergic oscillations actually decrease excitability and desynchronize firing. Tonic oscillations that are synaptically GABAergic-like those in real hippocampus-don't provoke a homeostatic response, however. If our simple model is correct, homeostasis can explain why the theta rhythm in the hippocampus is synaptically inhibitory: GABA has little to no intrinsic homeostatic response, and so can preserve the pyramidal cell's natural dynamic range. Based on these results we can also speculate that homeostasis may explain why AMPAergic oscillations in cortex, and hippocampus, often appear as bursts. Bursts do not interact with the slow homeostatic time constant, and so retain their normal excitatory effect."
8.  How adaptation makes low firing rates robust (Sherman & Ha 2017)
"Low frequency firing is modeled by Type 1 neurons with a SNIC (saddle node on an invariant circle), but, because of the vertical slope of the square-root-like f–I curve, low f only occurs over a narrow range of I. When an adaptive current is added, however, the f–I curve is linearized, and low f occurs robustly over a large I range. Ermentrout (Neural Comput. 10(7):1721-1729, 1998) showed that this feature of adaptation paradoxically arises from the SNIC that is responsible for the vertical slope. We show, using a simplified Hindmarsh–Rose neuron with negative feedback acting directly on the adaptation current, that whereas a SNIC contributes to linearization, in practice linearization over a large interval may require strong adaptation strength. We also find that a type 2 neuron with threshold generated by a Hopf bifurcation can also show linearization if adaptation strength is strong. Thus, a SNIC is not necessary. More fundamental than a SNIC is stretching the steep region near threshold, which stems from sufficiently strong adaptation, though a SNIC contributes if present. In a more realistic conductance-based model, Morris–Lecar, with negative feedback acting on the adaptation conductance, an additional assumption that the driving force of the adaptation current is independent of I is needed. If this holds, strong adaptive conductance is both necessary and sufficient for linearization of f–I curves of type 2 f–I curves."
9.  Loss of phase-locking in non-weakly coupled inhib. networks of type-I neurons (Oh and Matveev 2009)
... Here we examine the loss of synchrony caused by an increase in inhibitory coupling in networks of type-I Morris–Lecar model oscillators, which is characterized by a period-doubling cascade and leads to mode-locked states with alternation in the firing order of the two cells, as reported recently by Maran and Canavier (J Comput Nerosci, 2008) for a network of Wang-Buzsáki model neurons. Although alternating-order firing has been previously reported as a near-synchronous state, we show that the stable phase difference between the spikes of the two Morris–Lecar cells can constitute as much as 70% of the unperturbed oscillation period. Further, we examine the generality of this phenomenon for a class of type-I oscillators that are close to their excitation thresholds, and provide an intuitive geometric description of such “leap-frog” dynamics. ..."
10.  Modeling epileptic seizure induced by depolarization block (Kim & Dykamp 2017)
"The inhibitory restraint necessary to suppress aberrant activity can fail when inhibitory neurons cease to generate action potentials as they enter depolarization block. We investigate possible bifurcation structures that arise at the onset of seizure-like activity resulting from depolarization block in inhibitory neurons. Networks of conductance based excitatory and inhibitory neurons are simulated to characterize different types of transitions to the seizure state, and a mean field model is developed to verify the generality of the observed phenomena of excitatory-inhibitory dynamics. ..."
11.  Morris-Lecar model of the barnacle giant muscle fiber (Morris, Lecar 1981)
... This paper presents an analysis of the possible modes of behavior available to a system of two noninactivating conductance mechanisms, and indicates a good correspondence to the types of behavior exhibited by barnacle fiber. The differential equations of a simple equivalent circuit for the fiber are dealt with by means of some of the mathematical techniques of nonlinear mechanics. General features of the system are (a) a propensity to produce damped or sustained oscillations over a rather broad parameter range, and (b) considerable latitude in the shape of the oscillatory potentials. It is concluded that for cells subject to changeable parameters (either from cell to cell or with time during cellular activity), a system dominated by two noninactivating conductances can exhibit varied oscillatory and bistable behavior. See paper for details.
12.  Multiscale modeling of epileptic seizures (Naze et al. 2015)
" ... In the context of epilepsy, the functional properties of the network at the source of a seizure are disrupted by a possibly large set of factors at the cellular and molecular levels. It is therefore needed to sacrifice some biological accuracy to model seizure dynamics in favor of macroscopic realizations. Here, we present a neuronal network model that convenes both neuronal and network representations with the goal to describe brain dynamics involved in the development of epilepsy. We compare our modeling results with animal in vivo recordings to validate our approach in the context of seizures. ..."
13.  Origin of heterogeneous spiking patterns in spinal dorsal horn neurons (Balachandar & Prescott 2018)
"Neurons are often classified by spiking pattern. Yet, some neurons exhibit distinct patterns under subtly different test conditions, which suggests that they operate near an abrupt transition, or bifurcation. A set of such neurons may exhibit heterogeneous spiking patterns not because of qualitative differences in which ion channels they express, but rather because quantitative differences in expression levels cause neurons to operate on opposite sides of a bifurcation. Neurons in the spinal dorsal horn, for example, respond to somatic current injection with patterns that include tonic, single, gap, delayed and reluctant spiking. It is unclear whether these patterns reflect five cell populations (defined by distinct ion channel expression patterns), heterogeneity within a single population, or some combination thereof. We reproduced all five spiking patterns in a computational model by varying the densities of a low-threshold (KV1-type) potassium conductance and an inactivating (A-type) potassium conductance and found that single, gap, delayed and reluctant spiking arise when the joint probability distribution of those channel densities spans two intersecting bifurcations that divide the parameter space into quadrants, each associated with a different spiking pattern. ... "
14.  Pyramidal neurons switch from integrators to resonators (Prescott et al. 2008)
During wakefulness, pyramidal neurons in the intact brain are bombarded by synaptic input that causes tonic depolarization, increased membrane conductance (i.e. shunting), and noisy fluctuations in membrane potential; by comparison, pyramidal neurons in acute slices typically experience little background input. Such differences in operating conditions can compromise extrapolation of in vitro data to explain neuronal operation in vivo. ... in slice experiments, we show that CA1 hippocampal pyramidal cells switch from integrators to resonators, i.e. from class 1 to class 2 excitability. The switch is explained by increased outward current contributed by the M-type potassium current IM ... Thus, even so-called “intrinsic” properties may differ qualitatively between in vitro and in vivo conditions.
15.  Reliability of Morris-Lecar neurons with added T, h, and AHP currents (Zeldenrust et al. 2013)
We investigated the reliability of the timing of spikes in a spike train in a Morris-Lecar model with several extensions. A frozen Gaussian noise current, superimposed on a DC current, was injected. The neuron responded with spike trains that showed trial-to-trial variability. The reliability depends on the shape (steepness) of the current input versus spike frequency output curve. The model also allowed to study the contribution of three relevant ionic membrane currents to reliability: a T-type calcium current, a cation selective h-current and a calcium dependent potassium current in order to allow bursting, investigate the consequences of a more complex current-frequency relation and produce realistic firing rates.
16.  Reverberatory bursts propagation and synchronization in developing cultured NNs (Huang et al 2016)
"Developing networks of neural systems can exhibit spontaneous, synchronous activities called neural bursts, which can be important in the organization of functional neural circuits. ... Using a propagation model we infer the spreading speed of the spiking activity, which increases as the culture ages. We perform computer simulations of the system using a physiological model of spiking networks in two spatial dimensions and find the parameters that reproduce the observed resynchronization of spiking in the bursts. An analysis of the simulated dynamics suggests that the depletion of synaptic resources causes the resynchronization. The spatial propagation dynamics of the simulations match well with observations over the course of a burst and point to an interplay of the synaptic efficacy and the noisy neural self-activation in producing the morphology of the bursts."
17.  The activity phase of postsynaptic neurons (Bose et al 2004)
We show, in a simplified network consisting of an oscillator inhibiting a follower neuron, how the interaction between synaptic depression and a transient potassium current in the follower neuron determines the activity phase of this neuron. We derive a mathematical expression to determine at what phase of the oscillation the follower neuron becomes active. This expression can be used to understand which parameters determine the phase of activity of the follower as the frequency of the oscillator is changed. See paper for more.

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