LASCON 2020 The HodgkinHuxley Model Volker Steuber Biocomputation

LASCON 2020 The Hodgkin-Huxley Model Volker Steuber Biocomputation Research Group University of Hertfordshire UK v. steuber@herts. ac. uk

Brief introduction to compartmental modelling • Represent neuronal morphology by a number of interconnected compartments. • Each compartment contains a set of ion channel models that represent the channels at the corresponding location in the real cell. Pyramidal cell and simplified compartmental model.

Compartmental modelling • Use compartments that are small enough to be iso-potential. • Models of large neurons require large numbers of compartments. Cerebellar Purkinje cell and complex model with 4550 compartments (De Schutter & Bower 1994, imported into neuro. Construct).

Modelling a single compartment: equivalent circuit Membrane acts as a resistor in parallel with a capacitor. - lipid bilayer + + + ion channels

Simple rules for circuit analysis • Current through resistor is described by Ohm’s law: • Capacitive current: • Kirchhoff’s law: total sum of currents into a point equals total sum of currents out. In the absence of injected current: • Change of voltage: • Membrane time constant:

Voltage response to current injection Inject current steps into a cell (example Paramecium): • Can extract the membrane time constant tm • Cm is often around 1 m. F/cm 2 → can estimate Rm= tm/Cm

Current – voltage relation of ion channels Concentration gradients cause deviation from Ohm’s law. gs channel conductance, V – Es driving force, Es equilibrium or reversal potential Nernst equation: [S]out and [S]in extracellular and intracellular ion concentration, R Gas constant, F Faraday constant, T temperature, z valency (1 for K+, 2 for Ca 2+, -1 for Cl-).

Ion concentrations and reversal potentials For Mammalian skeletal muscle at 37˚C (varies depending on preparation and age, but for most neurons in the same range).

Updated equivalent electrical circuit of a compartment conductance battery capacitive current = leakage current + Na current + K current

Equivalent electrical circuit of a neuron v v

Axial currents flow between neighbouring compartments V’ v V R a’ V’’ Ra’’ capacitive current = channel currents + axial currents + injected current

Current – voltage relation of ion channels revisited Conductance is a function of voltage → non-linear I-V relationships gs channel conductance, V – Es driving force, Es equilibrium or reversal potential

Resting potential and action potential V Na+ K+ Membrane at rest is predominantly permeable to K → Goldman-Hodgkin-Katz voltage equation: PS membrane permeability of ion S V ≈ EK

Propagation of action potential along axon Cole & Curtis (1938 -1949), Hodgkin, Huxley & Katz (1937 -1952): classical studies of the squid giant axon.

Reminder: reversal potentials

Action potential is Na dependent Hodgkin & Huxley (1949): decreased extracellular [Na+] leads to reduced and delayed action potentials.

A new technique: voltage clamp Marmont (1949), Cole (1949), Hodgkin, Huxley & Katz (1949) silver wire squid axon

Measuring channel currents in voltage clamp Hodgkin & Huxley (1952) silver wire

Separation of Na and K currents Can isolate K current by replacing Na in the extracellular medium. Subtraction gives Na current.

Current – voltage relationships for Na and K

Time course of the Na and K conductance at different V Calculate conductances g. S from IS = g. S (V – ES). slow activation fast activation slow inactivation

The Hodgkin-Huxley model: fitting the K channel activation follows S-shaped time course → controlled by several independent processes or particles? voltage conductance Curve fitting suggests four independent “gating particles” n. constant maximum conductance time and voltage dependent activation variable Activation variable represents probability that gate is in permissive / open state (0 ≤ n ≤ 1).

K channel activation Transition between permissive and non-permissive state is a first-order reaction with rate constants an and bn : an 1 -n n bn Rate of change can be expressed as function of rate constants or as function of time constant tn and steady state value n∞:

Na channel activation and inactivation: two distinct gates Na channel activation also shows S-shaped time course (but less pronounced). New feature: inactivation. voltage conductance Need activation gate m and inactivation gate h: h is the probability that a channel is not inactivated.

Na channel activation and inactivation

Time course of activation and inactivation

Hodgkin-Huxley model predicts realistic action potentials Vm m 3 h n 4

Temperature dependence • Hodgkin et al. (1952): warming a squid axon by 10˚C speeds up activation and inactivation rates by a factor Q 10 ≈ 2 -4. • However, this varies between different channel types and some species express other channel types when exposed to different temperatures! • Channel conductance is less temperature sensitive (Q 10 ≈ 1. 2 -1. 5).

References Principles of Computational Modelling in Neuroscience. D. Sterratt, B. Graham, A. Gillies, D. Willshaw, Cambridge University Press (2011). Methods in Neuronal Modeling: from Ions to Networks. C. Koch and I. Segev eds. , MIT Press (1998). Computational Neuroscience: Realistic Modeling for Experimentalists. E. De Schutter editor, CRC Press (2000). Ion Channels of Excitable Membranes. B. Hille, Sinauer (2001). Foundations of Cellular Neurophysiology. D. Johnston and S. Wu, MIT Press (1994).