A Mean-Field Firing-Rate Model for the Suprachiasmatic Nucleus

We present a mean-field formalism for modeling firing-rate statistics of brain regions whose neurons exhibit atypical firing patterns and heterogeneous electrophysiological properties. We apply the formalism to the suprachiasmatic nucleus (SCN)---the human circadian pacemaker---whose neurons can intrinsically exhibit depolarized low-amplitude membrane oscillations (DLAMOs), depolariza-tion block (DB), and standard action potential firing at different times of day. Further, gamma-aminobutyric acid reversal potentials and molecular circadian phases of SCN neurons, among other properties, vary across the network and/or slowly over time. Our formalism consists of a system of integro-differential equations describing the time evolution of the mean and standard deviation of synaptic conductances across the network. Electrophysiological properties of SCN neurons are in-corporated by computing responses to synaptic conductance inputs of a Hodgkin-Huxley-type SCN neuron model that exhibits DLAMOs and DB. Such responses are then averaged over distributions of relevant quantities and included in the differential equations. Results suggest mechanisms by which physiologically relevant changes to firing activities may arise, highlighting means by which the amplitude of firing rates may shrink, the standard deviation of firing rates may grow, and by which a mid-day dip in firing rates may appear. For instance, results show that a large spread in circadian phases across SCN neurons reduces the size of oscillations in SCN network firing activity across the 24-hour day, identifying a mechanism by which heterogeneities in neuron electrophysiology could influence circadian rhythms.