*Principal investigator(s): Boris Gutkin, Martin Krupa (INRIA), Anatoly Buchin, Alexandre Hyafil, Lorenzo Fontolan (U Geneva)*

For the analysis of single neurons we take the minimal modeling approach, which attempts to construct the simplest structural model for explaining an observed phenomenon, using mechanisms based on known physiological processes. The goal of this approach is to create models that display non-trivial behavior, provide experimentally testable predictions, and are amenable to analysis. A central interest is developing mathematical methods that produce minimal models. In other words models that are mathematically tractable, and yet reflect the very heart of the neural mechanism that underlies the neural or cognitive phenomenon. This, slightly abstracted and mathematical approach can produce not only individual models, but also a theory with a wider brush stroke, and hopefully allow one to unify previously disparate data.

Recently we turned to modelling network activity during gamma oscillations, their coupling with the theta rhythm and also development of epileptic seizures (in collaboration with R Miles). For example, the transition from evoked to repetitive firing in many conductance-based neural models is produced by a specific dynamical mechanism: the saddle-node bifurcation, seen when the associated phase-space diagrams are studied in the context of non-linear dynamical systems. This is known as "Type I" membrane excitability, as defined by A. Hogdkin in his seminal work from 1948. By using normal form reduction a simple canonical one-equation model for this dynamical class can be derived and used to make qualitative statements generic to the whole class. Thus by studying this canonical model we are studying behavior of a wide class of excitable membranes and thus - neurons.

Neurons spike generating dynamics can be characterized by its Phase Response Function - a measure of how timing of individual spikes is shifted by weak transient inputs (e.g. single EPSP and/or IPSP). We are building upon the canonical theory of spike generation to predict the shapes of PRCs in cortical neurons and link these with other standard measures of neural response.

We have derived an extended reduced model that includes slow adaptation (spike frequency adaptation). We showed how such adaptation can shape the bifurcations underlying the generation of action potentials. Such adaptive processes, linked with a family of slow potassium currents, are under exquisite control of neuromdulators such as achetylcholine and dopamen. We are currently studying experimentally the effect of such modulators on the structure of the spike generating dynamics in cortical neurons.

We have identified that neurons can respond to transient excitatory inputs either with high precision, independent of the spike emission probability, or with variable (arbitrary) delays. In the second case the distribution of spike times depends on the spike emission probability. We are now identifying the mechanisms underlying this and studying its consequences for coding.

We are further exploring the consequence of spike generating dynamics and adaptation on the computation performed by the neurons in the context of Bayesian information processing.

**Publications**

Muller, L., Brette, R., and Gutkin, B.S., *Spike-timing dependent plasticity and feed-forward input oscillations produce precise and invariant spike phase-locking*, Frontiers in Neuroscience, in press, (2011).

Stiefel, K.M., Gutkin, B.S., and Sejnowski, T.E., *The effects of cholinergic neuromodulation on neuronal phase-response curves of modeled cortical neurons*, J Comput Neurosci, 29.2, 289-301 (2009).

Gutkin, B.S., Tuckwell, H., and Jost, J., *Random perturbations of spiking activity in a pair of coupled neurons*, Theory in the Biosciences, (in press), (2008).

Stiefel, K.M., Gutkin, B., and TE, T.E.S., *Cholinergic modulation of dynamics underlying spike generation in cortical neurons*, PLoS ONE, 3(12), e3947 (2008).

Jeong, H.Y. and Gutkin, B.S., *Synchrony of neuronal oscillations controlled by GABAergic reversal potentials*, Neural Computation, 19 (3), 706-729 (2007).

Brumberg, J.C. and Gutkin, B.S., *Cortical pyramidal cells as non-linear oscillators: Experiment and spike-generation theory*, Brain Research, 1171, 122-137 (2007).

Gutkin, B.S. and Ermentrout, G.B., *Neuroscience: spikes too kinky in the cortex?*, Nature, 440 (7087), 999-1000 (2006).

Gutkin, B.S., Ermentrout, G.B., and Reyes, A.D., *Phase-response curves give the responses of neurons to transient inputs.*, Journal of Neurophysiology, 94, 1623-1635 (2005).

Stiefel, K.M., Wespatat, V., Gutkin, B.S., Tennigkeit, F., and Singer, W., *Phase dependent sign changes of GABAergic synaptic input explored in vitro and in computo*, Journal of Computational Neuroscience, 19 (1), 71-85 (2005).

Gutkin, B.S., Ermentrout, G.B., and Rudolph, M., *Spike generating dynamics and the conditions for spike-time precision in cortical neurons*, Journal of Computational Neuroscience, 15, 91-103 (2003).

Ermentrout, G.B., Pascal, M., and Gutkin, B.S., *The Effects of Spike Frequency Adaptation and Negative Feedback on the Synchronization of Neural Oscillators*, Neural Computation, 13, 1285-1310 (2001).

Gutkin, B.S. and Ermentrout, G.B., *Dynamics of Membrane Excitability Determine Interspike Interval Variability: A Link between Spike Generation Mechanisms and Cortical Spike Train Statistics*, Neural Computation, 10 (5), 1047-1065 (1998).