Neuronal ion-channel dynamics in silicon

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Abstract
We present a simple silicon circuit for modelling voltage- dependent ion channels found within neural cells, capturing both the gating particle's sigmoidal activation (or inactivation) and the bell- shaped time constant. In its simplest form, our ion-channel analog consists of two MOS transistors and a unity-gain inverter. We present equations describing its nonlinear dynamics and measurements from a chip fabricated in a 0.25µm CMOS process. The channel analog's simplicity allows tens of thousands to be built on a single chip, facilitating the implementation of biologically realistic models of neural computation. I. ION CHANNELS The brain is one of the most powerful computing machines today, capable of outperforming modern machines in many computational tasks. As such, the field of neuromorphic engineering seeks to emulate the brain by using the transistor's physical properties to create silicon analogs of neural circuits. Silicon is an attractive medium because a single chip has thousands of (heterogeneous) silicon neurons that operate in real-time. One of the key computational components within the brain is the voltage-gated ion channel. Ion channels are pores within the mem- brane of neurons, gating the flow of ions between the intracellular and extracellular media. The gating dynamics are membrane-voltage- dependent, and once open, the ionic currents can greatly influence the output of the cell, either facilitating or hindering the generation of an action potential. To date, neuromorphicmodels have not captured the voltage depen- dence of the ion channel's temporal dynamics. Since neuromorphic circuits are constrained by surface area on the silicon die, larger, more complicated, circuits translate to fewer silicon neurons on a single chip. Thus, previous models of voltage-dependent ion channels (1), (2) have sacrificed the voltage dependenceof the time constant, opting to fix it to reduce the circuit size. In certain areas of the brain, however, this voltage dependence is critical. One example is the low-threshold calcium channel in the relay neurons of the thalamus: The time constant for inactivation can vary over an order of magnitude depending on the membrane voltage. This variation defines the relative lengths of the interburst interval (long) and the burst duration (short) when the cell bursts rhythmically. In this paper, we present a compact circuit that models the nonlin- ear dynamics of the ion channel's gating particles. Our circuit is based on linear thermodynamic models of ion channels (3), which apply thermodynamic considerations to the gating particle's movement. Similar considerations of the transistor makes clear that both the ion channel and the transistor operate under similar principles. This insight allows us to implement the voltage-dependence of the ion channel's temporal dynamics, while at the same time using fewer transistors than previous neuromorphic models that do not possess these nonlinear dynamics. With a more compact design, we can incorporate a larger number of silicon neurons on a chip without sacrificing biological realism.