# Mathematics Colloquia and Seminars

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### Non-smooth dynamics in spiking neural models

**Mathematical Biology**

Speaker: | Kyle Wedgwood, University of Nottingham, UK |

Location: | 2112 MSB |

Start time: | Mon, Nov 25 2013, 3:10PM |

An alternative to the construction of detailed biophysical models has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies. In this case, the single neuron model of choice is often a variant of the classic integrate-and-fire model, which is described by a non-smooth dynamical system. In this talk, I will review some of the basic behaviour of this class of models, and discuss the non-smooth effects of including a reset mechanism and its relevance to neuroscience. I will present a tractable planar single neuron model that captures a variety of the rich behaviour that neurons can exhibit. The techniques and terminology of non-smooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model, as well as to develop the notion of Lyapunov exponents. I also show how to construct the phase response curve for this system, emphasising that techniques in mathematical neuroscience may also translate back to the field of non-smooth dynamical systems. The stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times. At the network level, I will consider linear coupling between voltage variables, as would occur in neurobiological networks with gap-junction coupling, and show how to analyse the existence and stability of both the asynchronous and synchronous states. In the former case I will use a phase-density technique that is valid for any large system of globally coupled limit cycle oscillators, whilst in the latter I will use a novel technique that can handle the non-smooth reset of the model upon spiking.