**A sequence of Reductions in Mathematical Models of Primary Visual Cortex**
**Chizhov A.V.**
* A.F.Ioffe Physical-Technical Institute of Rassian Academy of Sciences, St.-Petersburg, 194021, Russia*

anton.chizhov@mail.ioffe.ru

**Abstract.** A hierarchical sequence of primary visual cortex models is constructed, each model processing the information about visual stimulus orientation. The most detailed model is based on the Conductance-Based Refractory Density (CBRD) approach for a population of realistic, Hodgkin-Huxley-like neurons. Populations in a continuum distributed along the cortical surface interact by means of synaptic connections. A certain set of parameters of connections and stimulation according to the pinwheel structure provides the effect of orientation tuning for different levels of stimulus contrast. The model realistically reproduces the stationary and transient regimes of cortical activity. The reduction of the model includes: 1) adaptive two-compartmental excitatory neurons and non-adaptive one-compartmental interneurons are substituted by the leaked integrate-and-fire (LIF) neurons; 2) pinwheel-like architecture of orientational hypercolumns is approximated by a ring of visual stimulus intensity gradient orientations; 3) the second order synaptic kinetics is reduced to transient kinetics; 4) the consideration of excitatory and inhibitory populations is reduced to only one explicit population; 5) the CBRD approach for LIF-neurons is substituted by the Fokker-Planck (FP) equation for neuronal density in the membrane potential space; 6) the FP equation is approximated by the stationary rate-current-conductance (f-I-G) dependence, and 7) f-I-G-function is reduced to threshold-linear f-I-function. Finally, the reduction leads to the canonical firing-rate ring-model. The comparison reveals the distinct roles of assumptions made during the reduction procedure.

**Key words**: *visual cortex, Hodgkin-Huxley neuron, neuronal populations, Fokker-Planck equation, refractory-density approach.*