MECHANISTIC AND MATHEMATICAL ASPECTS OF THE CREATION OF A THETA RHYTHM 
(8 - 12 Hz) IN A MODEL OF THE HIPPOCAMPAL CA1 AREA


Gillies et al. [1] have experimentally shown the existence of oscillation in 
the theta frequency range (8-10 Hz) in slices of the CA1 hippocampal area 
of the brain {\it in vitro} when phasic excitation is blocked. In the presence
of phasic excitation, the predominant frequency is in the gamma range 
(~ 40 Hz). Two different types of inhibitory neurons are involved in the
mechanism of generating these rhythms: standard interneurons and, so-called, 
OLM cells; the latter are turned on by inhibition but push the cells toward
spiking. They also have longer lasting inhibitory postsynaptic potentials 
(IPSP) in comparison with the former. 

We present a biophysically-inspired mathematical model that successfully 
reproduces experimental findings. This model focuses on the activity of O-LM 
(O), cells producing slow IPSPs, and other inhibitory neurons (I), each 
modeled as a single compartment. In addition to standard Hodgkin-Huxley 
currents, a hyperpolarization-activated (Ih) current was used for the O cells; 
blockade of Ih has been shown to destroy the rhythmicity both experimentally 
and in simulations.

We explain by means of numerical and analytical techniques the mechanism by 
which coherent theta oscillations are created, due to the interaction of the 
I and O cells via the fast and slow inhibition; we emphasize the effect that 
I cells exert on O cells due to the presence of Ih.

In an attempt to deeply understand the dynamics of the network, we perform a 
"geometric asymptotic analysis". In particular, we show that for a single O 
cell the interspike interval may be divided in subintervals inside which
the dynamics can be described by lower dimensional systems with slower 
currents as modulators. This approach can be exploited in order to explain
the synchronization properties in larger networks. This is work in progress. 



[1] M. J. Gillies, R. D. Traub, F. E. N Le Bleau, C. H. Davis, T. Gloveli, 
E. H. Buhl, and M. A. Whittington. A model of atropine-resistant theta 
oscillations in rat hippocampal area CA1. J Physiol (Lond), 543:779--793, 2002.
  



Joint work with M. J. Gillies, C. D. Acker, J. A. White, E. H. Buhl, 
M. A. Whittington and N. Kopell.