{"title": "A Systematic Study of the Input/Output Properties of a 2 Compartment Model Neuron With Active Membranes", "book": "Advances in Neural Information Processing Systems", "page_first": 149, "page_last": 159, "abstract": null, "full_text": "A Systematic Study or the Input/Output Properties \n\n149 \n\nA Systematic Study of the Input/Output Properties \n\nof a 2 Compartment Model Neuron \n\nWith Active Membranes \n\nPaul Rhodes \n\nUniversity of California, San Diego \n\nABSTRACT \n\nThe input/output properties of a 2 compartment model neuron are systematically \nexplored. Taken from the work of MacGregor (MacGregor, 1987), the model neuron \ncompartments contain several active conductances, including a potassium conductance in \nthe dendritic compartment driven by the accumulation of intradendritic calcium. \nDynamics of the conductances and potentials are governed by a set of coupled first order \ndifferential equations which are integrated numerically. There are a set of 17 internal \nparameters to this model, specificying conductance rate constants, time constants, \nthresholds, etc. \n\nTo study parameter sensitivity, a set of trials were run in which the input driving the \nneuron is kept fixed while each internal parameter is varied with all others left fixed. \n\nTo study the input/output relation, the input to the dendrite (a square wave) was varied \n(in frequency and magnitude) while all internal parameters of the system were left flXed, \nand the resulting output firing rate and bursting rate was counted. \n\nThe input/output relation of the model neuron studied turns out to be much more \nsensitive to modulation of certain dendritic potassium current parameters than to \nplasticity of synapse efficacy per se (the amount of current influx due to synapse \nactivation). This would in turn suggest, as has been recently observed experimentally, \nthat the potassium current may be as or more important a focus of neural plasticity than \nsynaptic efficacy. \n\nINTRODUCTION \n\nIn order to model biologically realistic neural systems, we will ultimately be seeking to \nconstruct networks with thousands of neurons and millions of interconnections. It is \ntherefor desireable to employ basic units with sufficient computational simplicity to \nmake meaningful simulations tractable, yet with sufficient fidelity to biological neurons \nthat we may retain a hope of gleaning by these simulations something about the activity \ngoing on during biological information processing. \n\n\fISO \n\nRhodes \n\nThe types of neuron models employed in the computational neuroscience literature range \nfrom binary threshold units to sigmoid transfer functions to 1500 compartment neurons \nwith Hodgkin-Huxley kinetics for a whole set of active conductances and spines with \nrich internal structure. In principle, a model neuron's functional participation in the \noperation of a network may be fully characterized by a complete description of its \ntransfer function, or input-output relation. This relation would necessarily be \nparameterized by a host of internal variables (which would include conductance rate \nconstants and parameters defining the neuron's morphology) as well as a very rich space \ncharacterizing possible variations in input (including location of input in dentritic tree). \nIn learning to judge which structural elements of highly realistic models must be \npreserved and which may be simplified, one approach will be to test the degree to which \nthe input-output relation of the simplified neuron (given a physiologically relevant \nparameter range and input space) is sufficiently close to the input-output properties of \nthe highly realistic model. \n\nTo define 'sufficiently close', we will ultimately refer to the operation of the network as \na whole as follows: the transfer function of a simplified neuron model will be considered \n'sufficiently close' to a more realistic neuron model if a chosen information processing \ntask carried out by the overall network is performed by a network built up of the \nsimplified neurons in a manner close to that observed in a network of the more realistic \nneurons. \n\nWe propose to begin by exploring the input/output properties of a greatly simplified 2 \ncompartment model neuron with active conductances. Even in this very simple structure \nthere are many (17) internal parameters for things like time constants and activation rates \nof currents. We wish to understand the parameter sensitivity of this model system and \ncharacterize its input-output relation. \n\n1.0 DESCRIPTION OF THE MODEL NEURON \n\nTHE MODEL NEURON CONSISTS OF A SOMA WITH A VOLTAGE-GATED \nPOTASSIUM CONDUCTANCE AND A SINGLE COMPARTMENT DENDRITE \nWITH A VOLTAGE-GATED CALCIUM CONDUCTANCE AND A [CAl-GATED \nPOTASSIUM CONDUCTANCE \n\nWe will choose for this study a simple model neuron described by MacGregor (I987). It \npossesses a single compartment dendrite. This is viewed as a crude approximation to the \nlumped reduction of a dendritic tree. In this approximation, we are neglecting spatial and \ntemporal summing of individual synaptic EPSP's distributed over a dendritic tree, as well \nas the spatial and temporal dispersion (smearing) due to transmission to the soma. The \nindividual inputs we will be using are large enough to drive the soma to firing, and so \nwould represent the summation of many relatively simultaneous individual EPSPs, \nperhaps as from the set of contacts upon a neuron's dendritic tree made by the \narborization of one different axon. The dendritic membrane possesses a potassium \nconductance gated by intradendritic calcium concentration and a voltage gated calcium \nconductance. The soma contains its own voltage-gated potassium channels and \nmembrane time constants. Electrical connection between soma and dendrite is expressed \nby an input impedance in each direction. The soma fires an action potential, simply \nexpressed by raising its voltage to 50 mv for one msec after its internal voltage has been \n\n\fA Systematic Study or the Input/Output Properties \n\n151 \n\ndriven to firing threshold. Calcium accumulation in the dendrite is modelled assuming \naccumulation proportional to calcium conductance. Calcium conductance itself increases \nin proportion to the difference between the dendrite's voltage and a threshold, and calcium \nis removed from the dendrite by means of an exponential decay. This system is modelled \nby a set of coupled frrst order differential equations as follows: \n\n1.1 THE SET OF EQUATIONS GOVERNING THE DYNAMIC \nVARIABLES OF THIS MODEL \n\nThe soma's voltage ES is governed by: \n\ndES/dt={ -ES+SOMAINPUT +GDS *(ED-ES)+GKS * (EK-ES)} IfS \n\nwhere SOMAINPUT is obtained by dividing the input current by the total resting \nconductance of the dendrite (therefor it has units of voltage). GDS is proportional to \ninput resistance from dendrite to soma, and multiplies the difference between the \ndendrite's voltage ED and the soma's voltage ES; GKS is the soma's aggregate \npotassium conductance (modelled below); EK is the voltage of the potassium battery \n(assumed constant at -1 Omv); and TS is the soma's time constant. All potentials are \nrelative to resting potential, and all conductances are dimensionless. \n\nThe dendrite's voltage ED is govened by: \n\ndED/dt={-ED+DENDINPUT+GSD*(ES-ED)+GCA*(ECA-ED)+ GKD*(EK-ED)}IID \n\nwhere DENDINPUT is obtained by dividing the input current by the total resting \nconductance of the dendrite and so has units of voltage. GSD is proportional to the \ninput resistance from soma to dendrite, and hence multiplies the difference between ES \nand ED; GCA is the dendrite's calcium conductance (modelled below), ECA is the \ncalcium battery (assumed constant at 50mv), and GKD is proportional to the dendrite's \npotassium conductance (modelled below). All potentials are relative to resting potential. \n\nThe soma's voltage is raised artificially to 50mv for I msec after the soma's voltage \nexceeds a (fixed) threshold, thus simplifying the action potential. \n\nThe potassium conductance in the soma, GKS, is governed by: \n\ndGKS/dt={ -GKS+S*B}lfGK \n\nwhere S is 1 if an action potential has just fired and 0 otherwise, B is an activation rate \nconstant governing the rate of increase of potassium conductance, and TGK is the time \nconstant of the potassium conductance decay. This rather simplified picture of \npotassium conductance will be replaced by a more realistic version with a Markov state \nmodel of the potassium channel in a subsequent publication in preparation. For the \npresent investigation then we are modelling the voltage dependence of the potassium \nconductance by the following: potassium conductance builds up by a fixed amount \n(proportional to BlfGK) during each action potential, and thereafter decays exponentially \nwith time constant TGK. \n\n\f152 \n\nRhodes \n\nThe dendrite's calcium conductance is governed by: \n\ndGCNdt={ -GCA +D*(ED-CSPlKETHRESH)} IfGCA \ndGCNdt={ -GCNlGCA} \n\nED>CSPIKETHRESH \n\nEDCALCTHRESH \n[CA]