# Dynamical Systems: Overview

**DOI:**https://doi.org/10.1007/978-1-4614-7320-6_768-1

## Keywords

Neural Dynamic Computational Neuroscience Frequency Adaptation Fire Model Lower Dimensional Manifold## Definition

Many models of computational neuroscience are formulated in terms of nonlinear dynamical system (sometimes the dynamical system with noise) and include a large number of parameters. Therefore, it is difficult to analyze dynamics and find correspondence between parameter values and dynamical mode. Mathematical theory of dynamical systems and bifurcations provide a valuable tool for a qualitative study and finding regions in parameter space corresponding to different dynamical modes (e.g., oscillations or bistability). Thus, the dynamical systems (or nonlinear dynamics) approach to analysis of neural systems has played a central role for computational neuroscience for many years (summarized, e.g., in recent textbooks, Izhikevich (2007) and Ermentrout and Terman (2010)).

## Detailed Description

Theory of dynamical systems and bifurcations provides a list of universal scenarios of dynamics changes under parameter variation. For example, one scenario explaining the onset of oscillations is described by Andronov-Hopf bifurcation. In terms of neuroscience, this theory provides insight into the mechanisms underlying different neural response properties and firing patterns. Even more importantly, it allowed to elucidate the underlying mathematical structure that might be common to whole classes of firing behaviors. Thus, conclusions of such studies can be wide ranging, even if specific biophysical details of implementation may differ from case to case.

A common theme in all articles in this section is that they use concepts from nonlinear dynamics, especially geometrical methods like phase planes and bifurcation diagrams. Many exploit time scale differences to reduce dimensionality by dissecting the dynamics using fast-slow analysis, i.e., to separately understand the behaviors on the different time scales and then patch the behaviors together.

The articles in this section for the most part exploit the idealized model of neuron localized at one point (i.e., electrically compact neuron), focusing on the nonlinearities of spiking dynamics, and using biophysically minimal but biologically plausible description of neural dynamics. An article on “Fitzhugh-Nagumo model” describes one of the first examples where dynamical systems approach has been applied to analysis of neural dynamics (Fitzhugh 1955). Although “Fitzhugh-Nagumo Model” is not biologically grounded and formulated in terms of the cubic nonlinearity, the model is still considered as one of the prototype models for excitable systems.

An article on “Morris-Lecar model” describes the rich dynamic repertoire of the two-variable Morris-Lecar model, as its biophysical parameters are varied. The original detailed analysis of this model (Rinzel and Ermentrout 1998) has laid ground for similar approaches in many different contexts and provided a theoretical justification for influential Hodgkin’s classification of “excitability types” (Hodgkin 1948, described in “Excitability: Types I, II, and III”).

The articles on “Integrate and Fire Models, Deterministic” and “Theta Neuron Model” models target the most idealized end of the modelling spectrum where the spiking activity is represented by simple mathematical models. This approach to modelling of spiking times is fruitful for implementation in the large neural network and when mathematical analysis (in addition to numerical simulations) is desired.

Somewhat more intricate features of single cell dynamics are described in articles on modelling of “Post Inhibitory Rebound and Facilitation” and “Spike Frequency Adaptation” phenomena. Finally, the fast-slow dissection and bifurcation analysis really shine in the description of bursting behavior. The bursting dynamics (of individual cells or of population activity) is dissected into active and silent phases when trajectories are restricted to lower dimensional manifolds, and transitions between these phases correspond to reaching the manifold’s boundary and jumping to a different manifold.

While most of these examples are for single-cell dynamics, the qualitative mathematical study is also applicable to the dynamics of neuronal networks and structures, especially in the mean-field approximations. One such example for network-generated rhythms is presented in the article on “firing rate adaptation”.

Of course, the models presented in this section are rather idealized and simplified for mathematical study compared to many other neuronal models that are designed to investigate the biophysical details of action potential generation: interaction of many known ionic currents, or the spatial propagation of activity, etc. Such minimalistic models however are invaluable when the problem or question at hand require only qualitative or semiquantitative characterizations of spiking activity. This is especially important in studies of large networks of interacting cells.

## Cross-References

## References

- Ermentrout GB, Terman D (2010) Mathematical foundations of neuroscience. Springer, New YorkCrossRefGoogle Scholar
- Fitzhugh R (1955) Mathematical models of threshold phenomena in the nerve membrane. Bull Math Biophys 17(4):257–278CrossRefGoogle Scholar
- Hodgkin AL (1948) The local electric changes associated with repetitive action in a non-medullated axon. J Physiol Lond 107:165–181PubMedCentralPubMedGoogle Scholar
- Izhikevich EM (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. MIT Press, Cambridge, MAGoogle Scholar
- Rinzel J, Ermentrout B (1998) Analysis of neural excitability and oscillations. In: Koch C, Segev I (eds) Methods in neuronal modeling: from ions to networks, 2nd edn. MIT Press, Cambridge, MA, pp 251–291Google Scholar