In this part of the Supplementary Information we introduce a model of AP initiation by cooperative activation of voltage-gated sodium channels and characterize its basic properties. Then we describe the computational consequences of the characteristic features of cortical action potential initiation. Using a novel phenomenological neuron model, we show that these features allow a neuronal population to encode rapidly varying signals and to suppress responses to slowly varying stimuli. Contents AP initiation with cooperative sodium channel activation 2The single channel model 4Mean field model of cooperative gating 5The collective sodium activation curve 7AP generation with cooperative sodium channels 8Rate functions and parameters 8Functional consequences of 'anomalous' AP initiation 10 Modelling inter-channel cooperativity: In the coupled model, the opening of neighbouring channels shifts the single channel activation curve to more hyperpolarized potentials such that the probability of channel opening at a given MP is increased. AP initiation with cooperative sodium channel activation The single channel modelOur model of cooperative sodium channel activation is based on a single channel model originally introduced by Aldrich, Corey and Stevens (1983). Its state transition scheme (Fig. 3SI a) (Fig. 3SI a). The dynamics of a population of such channels is described by kinetic equations for the fraction of open channels and the fraction of available channelsFor discussing model properties it is useful to consider two characteristic functions:The instantaneous single channel activation curve, and the equilibrium inactivation functionIf the time scale of activation is shorter than the inactivation time constant I τ , the activation of channels from the available fraction, ( ) H t , is described by the instantaneous single channel activation curve (13) .For constant membrane potential and negligible sodium channel activation, the equilibrium inactivation function (14) ( )describes the equilibrium fraction of inactivated channels as a function of membrane potential. 5 Mean field model of cooperative gatingBased on the above model of single channel gating we formulated a mean field model of cooperative sodium channel activation. To derive this model, we assumed that each channel is coupled to K neighbouring channels. Opening of any of these neighbours is then assumed to cause a shift of the instantaneous activation curve of the channel by towards lower membrane potentials. The activation and deactivation rates of channel are then, O t V KJO t H t O t V KJO t O t H t V H t V H t O twhere is the instantaneous activation curve of independent channels. This equation is a self-consistency relation for . In the uncoupled case, , the collective sodium activation curve equals the single channel activation curve. For , the collective sodium activation curve becomes progressively steeper than with increasing values of and develops a discontinuous jump at a critical potential when the coupling strength becomes larger than a c...

The brain’s visual cortex processes information concerning form, pattern, and motion within functional maps that reflect the layout of neuronal circuits. We analyzed functional maps of orientation preference in the ferret, tree shrew, and galago—three species separated since the basal radiation of placental mammals more than 65 million years ago—and found a common organizing principle. A symmetry-based class of models for the self-organization of cortical networks predicts all essential features of the layout of these neuronal circuits, but only if suppressive long-range interactions dominate development. We show mathematically that orientation-selective long-range connectivity can mediate the required interactions. Our results suggest that self-organization has canalized the evolution of the neuronal circuitry underlying orientation preference maps into a single common design.