# BCCN Seminar

Thursday, 30.06.2005 17 c.t.## On the canonical theory of spike generation dynamics in cortical neurons

*by Dr. Boris Gutkin*

Contact person: Björn Naundorf

### Location

Seminarraum Haus 2, 4. Stock (Bunsenstr.)### Abstract

Neurons produce their activity (single action potentials and/or repetitive firing) due to recurrent interplay of intrinsic voltage dependent conductances. Thus a neuron may be considered as a complex dynamical system that undergoes input- and state-dependent bifurcations. We shall show that taking such dynamical systems view of neural excitability can lead to a theory of spike generating dynamics based on a classical classification scheme and yielding a simple canonical model. Vast majority of regular firing (non bursting) neurons can be put into two broad classes of excitability. Type II is characterised by non-zero minimal firing frequency, graded action potentials and a presence of sub-threshold resonances. Squid axon is a classical example of such, and the Hodgkin-Huxley model shows such behaviour. Type I membranes are characterized by arbitrarily low firing frequency, all-or-none spikes, and a general absence of resonances. Regular firing pyramidal neurons are an example of this type. The two types of excitability are linked to different bifurcations leading to the onset of repetitive firing: subcritical Andronov-Hopf for type II and a SNIC (saddle-node on an invariant circle) for type I. I shall concentrate on the latter and show how a canonical 1-dimensional model ( quadratic integrate-and-fire or theta-neuron) can be deduced for this class of neurons. I will then show how the model can be extended to account for slow modulatory processes in a mathematically coherent manner. I will then proceed to explain how slow adaptation can modify the bifurcation structure of the extended canonical model by moving it from type I to type II regime. Thus the extended theta-neuron can server as a canonical model for both types of spike-generating dynamics. In the second part of the talk I will show experimental data supporting the theory. In particular, I look for signatures of the bifurcation structure and its consequence. In general bifurcation are difficult to pin-down directly in biological systems, yet one can observe particular quantities uniquely associated with each type of spike generating dynamics. One such quantity is the Phase Response Curve (PRC). The PRC measures how the firing phase of a cell is perturbed by a weak transient input (e.g a synapse). Type I dynamics imply a strictly positive PRC, while type II, PRCs with negative regions. I shall show how slow adaptation changes the shape of the PRC, making it into type II. I will then show data from in vitro recordings that corroborate such a shift by neuromodulators that block adaptation. I will also discuss the notion of the firing threshold as it is defined by the type I dynamics, and show recent in vitro data suggesting that cortical pyramidal neurons exhibit precisely the threshold behavior consistent with a SNIC.