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The mammalian hippocampus forms a cognitive map using neurons that fire according to an animal's position ("place cells") and many other behavioral and cognitive variables. The responses of these neurons are shaped by their presynaptic inputs and the nature of their postsynaptic integration. In CA1 pyramidal neurons, spatial responses in vivo exhibit a strikingly supralinear dependence on baseline membrane potential. The biophysical mechanisms underlying this nonlinear cellular computation are unknown. Here, through a combination of in vitro, in vivo, and in silico approaches, we show that persistent sodium current mediates the strong membrane potential dependence of place cell activity. This current operates at membrane potentials below the action potential threshold and over seconds-long timescales, mediating a powerful and rapidly reversible amplification of synaptic responses, which drives place cell firing. Thus, we identify a biophysical mechanism that shapes the coding properties of neurons composing the hippocampal cognitive map.