Modelling Neurons Computing and the Brain Compartmental Modelling Spring 2010 2 1
Equivalent Electrical Circuit A patch of membrane is equivalent to an electrical circuit This circuit can be described by a differential equation: (Single Neuron Computation, pg 88) Spring 2010 3 Electrical Circuit along a Neurite Electrical circuit for a neurite (dendrite or axon) or the entire neuron obtained by joining equivalent circuits together with resistances Coupling resistance (r i or R a ) models the intracellular resistance of neuron (JW95, Fig 6.14) Modelled by an extension to the differential equation (The Book of Spring 2010 GENESIS, pg 8) 4 2
Compartmental Model of a Neuron Each compartment modelled by an equivalent circuit Compartments are chosen to be small enough that the membrane potential can be considered the same over that patch of membrane Spring 2010 5 Detailed Compartmental Model Lots of compartments are used that map closely to the morphology (shape) of the neuron Gives accurate model of voltage across the neuron BUT computationally intensive (Single Neuron Computation, pg 88) Spring 2010 6 3
Simplified Compartmental Model If modelling a network of neurons it may be necessary to reduce the number of compartments Aim to capture the basic morphology (shape) of the neuron (The Book of GENESIS, pg 8) Spring 2010 7 Large-scale Example Purkinje cell from cerebellum 4550 compartments 8021 ion channels 10 different voltage-dependent d t channels distributed across compartments 3500 synapses Reproduces electrical current injection and synaptic responses (De Schutter and Bower, 1994) Spring 2010 8 4
Modelling Ion Channel Currents Compartmental model requires electrical current, I X, through each type of ion channel: Need to model the changing conductance, G X Common approach based on the Hodgkin and Huxley (1952) model of the Na and K conductances underpinning the action potential Magnitude and rates of activation, inactivation and deactivation of the channels is a function of the membrane voltage Based on the concept of gates that control the opening and closing of the channels Spring 2010 9 (LK91, Fig 4.1) Modelling Ion Channel Currents Gate parameters: m, h, n Change over time as a function of membrane voltage, V E.g. m = f(v,t) (JW95, Fig. 6.9) Spring 2010 10 5
Modelling Synaptic Currents Synaptic conductances usually modelled as simple functions of time Conductance is zero until an AP arrives at the presynaptic terminal Then G syn evolves over time in response to neurotransmitter release E.g. G syn = g(t) Used to generate EPSPs and IPSPs Spring 2010 11 Simplified Models of Spiking Neurons Major property of biological neurons is firing of action potentials in response to synaptic input Essential features of this include: Action potential shape is same each time, so can just treat the occurrence of an AP (spike) as a binary 1 All EPSPs and IPSPs eventually get summed in the cell body, so do not need to model dendrites Integrate-and-fire model Point neuron Resistance + capacitance Synaptic input Spring 2010 12 6
Rate-based Neuron Models Computing units used in ANNs can be regarded as models of biological neurons Activation function captures essence of a neuron s spiking response f(x) Binary (0 or 1): neuron is silent or firing McCulloch-Pitts threshold logic unit Analog: number represents neuron firing rate Piece-wise linear or sigmoid f(x) x f(x) x x Spring 2010 13 7