# Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting

@inproceedings{Izhikevich2006DynamicalSI, title={Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting}, author={Eugene M. Izhikevich}, year={2006} }

This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition. In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics. This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students…

## 2,893 Citations

### The Dynamics of Neurons in a Minimal Model for Synaptic Integration

- Biology
- 2022

A simple model for the summation of excitatory inputs from synapses is studied and it is shown that a simple model can nonetheless highlight the general statistical features of spike timings in neural processes where the active properties of dendrites do not play a major role.

### The Complexity in Activity of Biological Neurons

- Biology
- 2011

This chapter shows many firing patterns in theoretical neuronal models or neurophysiological experiments of single neurons in rats, and depicts the phenomena of stochastic resonance and coherence resonance, and gives their dynamical mechanisms.

### Nonlinear and Stochastic Methods in Neurosciences

- Computer Science
- 2008

The problem of characterizing the probability distribution of spike timings can be reduced to the problem of first hitting times of certain stochastic process, and the derivation of mesoscopic description is presented, and it is proved that the equation obtained is well posed in the mathematical sense.

### Dynamical systems and their applications in neuroscience

- Biology
- 2007

It is shown that PRCs can be used to determine the synchronizing and/or phase-locking abilities of neural networks, and how the connection delay plays a role in this, and some phenomena to do with PRCs and bifurcations are demonstrated.

### Interplay between network topology and dynamics in neural systems

- Computer Science
- 2011

This thesis is a compendium of research which brings together ideas from the fields of Complex Networks and Computational Neuroscience to address two questions regarding neural systems: 1) How the…

### Nonlinear analysis methods in neural field models

- Biology
- 2011

This thesis deals with mesoscopic models of cortex called neural fields, and derives the main equations of Wilson and Cowan, which consist of integro-differential equations with delays modeling the signal propagation and the passage of signals across synapses and the dendritic tree.

### Neuronal Bifurcation Analysis and Construction of an Oscillatory Associative Memory

- Biology
- 2019

The first part of this project is to simulate neuronal dynamics using a quadratic integrate-and-fire model of the neuron that contains four dimensionless parameters (Izhikevich 2014), achieved by tuning the parameters in response to bifurcation analysis of phase portraits.

### Simulation of Large Scale Cortical Networks by Individual Neuron Dynamics

- Computer ScienceInt. J. Bifurc. Chaos
- 2010

It is found that replacing each cortical area by a single Rulkov map recreates the patterns of dynamical correlation of the multilevel model, while the outcome of other models and setups mainly depends on the local network properties.

### Chaotic neural circuit dynamics

- Biology
- 2017

Novel numerical and analytical techniques from dynamical systems, stochastic processes and information theory are developed to characterize the evoked and spontaneous dynamics and phase space organization of large neural circuit models to determine how biophysical properties of neurons and network parameters influence information transmission.

### Synchronization, Neuronal Excitability, and Information Flow in Networks of Neuronal Oscillators

- Biology
- 2012

A general model of pulse-coupled neuronal threshold units with a partial reset that captures the response of neurons to supra-threshold stimulation is proposed and it is shown that the excitability type of neurons, i.e. their intrinsic characteristic dynamics of generating action potentials, can be changed dynamically.

## References

SHOWING 1-10 OF 247 REFERENCES

### An introduction to the mathematics of neurons

- Computer Science
- 1986

The author's aim is to uncover frequency-response properties of neurons and to show that neural networks can support stable patterns of synchronized firing using a novel electrical circuit model of a neuron, called VCON, which shares many features with the Hodgkin-Huxley model, though it is much simpler to study.

### A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue

- BiologyKybernetik
- 2004

It is shown that this particular mode reproduces some of the phenomenology of visual psychophysics, including spatial modulation transfer function determinations, certain metacontrast effects, and the spatial hysteresis phenomenon found in stereopsis.

### Dissection of a model for neuronal parabolic bursting

- BiologyJournal of mathematical biology
- 1987

This work studies Plant's model for Aplysia R-15 to illustrate the view of these so-called “parabolic” bursters, and shows how bursting arises from an alternative mechanism in which calcium inactivates the calcium current and the potassium conductance is insensitive to calcium.

### Electrical Synapses and Synchrony: The Role of Intrinsic Currents

- BiologyThe Journal of Neuroscience
- 2003

Numerical simulations of a network of conductance-based neurons randomly connected with electrical synapses show that potassium currents promote synchrony, whereas the persistent sodium current impedes it and establish general rules to predict the dynamic state of large networks of neurons coupled with electricalsynapses.

### Dynamics of Strongly Coupled Spiking Neurons

- PhysicsNeural Computation
- 2000

It is shown how phase-locked states that are stable in the weak coupling regime can destabilize as the coupling is increased, leading to states characterized by spatiotemporal variations in the interspike intervals (ISIs).

### Asynchronous States and the Emergence of Synchrony in Large Networks of Interacting Excitatory and Inhibitory Neurons

- BiologyNeural Computation
- 2003

This work investigates theoretically the conditions for the emergence of synchronous activity in large networks, consisting of two populations of extensively connected neurons, one excitatory and one inhibitory, and shows that these mechanisms can be differentiated by the firing patterns they generate and their dependence on the mutual interactions of the inhibitory neurons and cross talk between the two populations.

### Analysis of an autonomous phase model for neuronal parabolic bursting

- Biology, PhysicsJournal of mathematical biology
- 1995

A simple phase model developed to study endogenous parabolic bursting, a class of burst activity observed experimentally in excitable membrane, demonstrates that it captures many dynamical features of more complex biophysical models.

### Properties of a Bursting Model with Two Slow Inhibitory Variables

- BiologySIAM J. Appl. Math.
- 1993

A geometric understanding of the solutionstructure and of transitions between various modes of behavior was developed and a novel use of the bifurcation code AUTO finds nullclines for the slow variables when the fast variables are periodic by averaging over the fast oscillations.

### Mechanism of bistability: tonic spiking and bursting in a neuron model.

- BiologyPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2005

It is argued that the Lukyanov-Shilnikov bifurcation of a saddle-node periodic orbit with noncentral homoclinics may initiate a bistability observed in a model of a leech heart interneuron under defined pharmacological conditions.

### A model of spindle rhythmicity in the isolated thalamic reticular nucleus.

- BiologyJournal of neurophysiology
- 1994

It is demonstrated that networks of model neurons that include the main intrinsic currents found in RE cells can generate waxing and waning oscillatory activity similar to the spindle rhythmicity observed in the isolated RE nucleus in vivo.