Electronic Model of Neural Adaptation E Paxon Frady Introduction The behavior of neurons has been modeled with circuit elements for a long time Hodgkin and Huxley developed their theory of action potential generation through circuit elements with active variable conductances that generated the non linear dynamics of the action potential Luckily for Hodgkin and Huxley the squid axon which they studied only contained two types of active channels It is known now that neurons can contain many different types of channels all of which can play a substantial role in the firing properties of the neuron One property of neuronal firing patterns that has attracted much attention recently is neural adaptation Adaptation was first observed when neurons would change their firing rates in the presence of a constant stimulus ref Originally it was thought that this process was mediated by a single exponential time constant a property which could easily be explained by a simple adaptation channel However evidence shows that the process of neural adaptation has different time constants dependent on the history of the input stimuli This history dependent modification of neuronal firing patterns could be a useful computational tool in the nervous system neuron firing rates are not based on local stimulus information but on history thus providing memory on a longer timescale Rather than exponentials evidence shows that neurons follow a power law based mechanism Several computational advantages such as time scale independence of power law dynamics are discussed in Drew and Abbott One mystery of power law adaptation dynamics is the implementation of such dynamics Several implementations have been suggested Drew and Abbott Fairhall Gilboa as the biophysical mechanism for power law adaptation Here we show a circuit model of neuronal dynamics based on Guy Roy s neuroFET with an additional adaptation channel that is time constant independent Methods Spiking Neuron Circuit Implementation The model of spiking neuron used is based on Guy Roy s neuroFET This model was developed as an implementation of Hodgkin and Huxley s description of a neuron and uses FET properties in the linear regime to act as the voltage dependent resistors that mimic channels To implement the dynamics of neuronal channels the voltage across the membrane capacitor is fed into wave form shaping circuits that act to mimic the dynamics of sodium and potassium channels These dynamics are fed into the FETs to alter their resistances and effectively open or close the channels Figure 1 The model neuron with only the two original channels behaved similarly to real neurons The resting potential was 45mV the peak of the AP was 30mV and the half width of the AP was 2 5 ms For a series of action potentials no changes in firing rate was observed after the first action potential The initial action potential was typically larger and broader than successive action potentials After each action potential there was a hyperpolarization of the voltage across the membrane capacitor that had a peak of 60 mV and decayed back to the resting potential with a time constant of approximately 10ms The formation of a series of action potentials appeared to follow a Hopf bifurcation Figure 1 General idea of circuit design using FETs in the linear regime to act as variable conductances Wave shaping network circuits are used to mimick the dynamics and voltage dependence of the channels Implementation of Adaptation In order to implement a time constant independent adaptation in the circuit we first created an additional channel to serve as the adaptation channel This channel was driven in a similar way by a wave shaping circuit To make the channel time constant independent the channel is itself driven by an RC circuit where R is in part determined by a FET acting as a variable resistor The FET is driven by an integrator circuit which keeps track of a long time range of the circuit history The change in the FET s conductance will change the R in the RC circuit leading to a change in the time constant of this circuit The RC circuit s output drives the FET channel that causes the neuron to adapt In order for the properties of the voltages driving the FET s conductances to be appropriate we had to amplify and bias these voltages so that they fall in the right range to alter the FET s resistances The amplification was done so that the voltage would fall in a dynamic range spanning between 5 and 10 volts and the bias was added so that this range would be mostly positive In order to fine tune the parameters variable voltage dividers were added as the bias parts of the Op Amp circuits and were set to find optimal ranges of the driving voltages The RC circuit that drives the adaptation channel originally used the amplified voltage as the source of the short term channel dynamics However this voltage is large and tended to saturate the transistor Instead a voltage divider was added to weaken the voltage across the source and drain of the transistor to keep it in the linear regime Later this voltage was amplified by another Op Amp circuit in order to return the voltage to a level that appropriately modifies the conductance of the FET regulating the adaptation channel Results After the new channel was implemented and the parameters of the circuit were tweaked the circuit showed adaptive properties There was a clear difference between the initial firing rate and the firing rate near the end of a stimulation The extent of adaptation could be varied by changing many of the parameters such as the bias of the voltage driving the adaptation channel the resistances that are in parallel and in series with the adaptation channel these are effectively the minimum and maximum Figure 2 Complete circuit diagram used to implement adaptation dynamics conductance values for the channel the bias of the integrator as well as the voltage driving the variable RC circuit Typically the circuit would quickly jump from the minimum and maximum adaptive rates leaving little in the duration of the more variable part of the adaptation The dynamic range of the adaptation times is likely to improve with a more refined parameter search of the resistances and capacitances that make up the adaptation circuit A further problem is the tendency for the adaptation channel to completely kill the generation of action potentials Increasing the strength of the adaptation channel by increasing the conductance in series