advertisement

Basic Models in Theoretical Neuroscience Oren Shriki 2010 Integrate and Fire and Conductance Based Neurons 1 2 References about neurons as electrical circuits: • Koch, C. Biophysics of Computation, Oxford Univ. Press, 1998. • Tuckwell, HC. Introduction to Theoretical Neurobiology, I&II, Cambridge UP, 1988. 3 The Neuron as an Electric Circuit 4 Intracellular Recording 5 Generation of Electric Potential on Nerve Cell Membranes Chief factors that determine the resting membrane potential: • The relative permeability of the membrane to different ions • Differences in ionic concentrations Ion pumps – Maintain the concentration gradient by actively moving ions against the gradient using metabolic resources. Ion channels – “Holes” that allow the passage of ions in the direction of the concentration gradient. Some channels are selective for specific ions and some are not selective. 6 Ion Channels and Ion Pumps 7 The Neuron as an Electric Circuit • Differences in ionic concentrations Battery • Cell membrane Capacitor • Ionic channels Resistors 8 The Neuron as an Electric Circuit Extracellular Intracellular 9 RC circuits Current source I C • R – Resistance (in Ohms) • C – Capacitance (in Farads) R 10 RC circuits I C R • The dynamical equation is: C dV dt V R I (t ) 11 RC circuits • Defining: • We obtain: RC dV V I ( t ) R dt • The general solution is: V t V 0 e t t R d t e t t I t 0 12 RC circuit • Response to a step current: t0 I I t 0 t dt e t t t0 I t e 0 t t t I dt e 0 e t t I ( e |0 e t V t V 0 e t t t I e 1 I t IR 1 e V t 1 e 13 RC circuit • Response to a step current: 14 The Integrate-and-Fire Neuron inside EL I C R Threshold mechanism outside • R – Membrane Resistance (1/conductance) • C – Membrane Capacitance (in Farads) 15 Integrate-and-Fire Neuron • If we define: V V in V out • The dynamical equation will be: dV 1 C V E L I ( t ) dt R • To simplify, we define: V V in V out E L • Thus: dV V C I (t ) 16 dt R Integrate-and-Fire Neuron • The threshold mechanism: – For V<θ the cell obeys its passive dynamics – For V=θ the cell fires a spike and the voltage resets to 0. • After voltage reset there is a refractory period, τR. 17 Integrate-and-Fire Neuron • Response to a step current: IR<θ: V t 18 Integrate-and-Fire Neuron • Response to a step current: IR>θ: V τR τR τR t T 19 Integrate-and-Fire Neuron • Finding the firing rate as a function of the applied current: V t IR 1 e 1 e T R T R e T R IR T R ln 1 IR T R ln 1 IR 1 IR T R ln 1 IR 20 Integrate-and-Fire Neuron 1 f I 1 T R ln 1 IR 1 R ln 1 IR 1 R C gL gL ln 1 I f 1 R Ic R I 21 The Hodgkin-Huxley Equations C dV dt I ion V , w I ( t ) I ion V , m , h , n g Na m h (V V Na ) g K n (V V K ) g L (V V L ) 3 4 dm/dt m (V)-m /τ m dh/dt h (V)-h /τ h dn/dt n (V)-n /τ n 22 The Hodgkin & Huxley Framework C dV dt I ion V , W 1 , , W N I ( t ) Each gating variable obeys the following dynamics: dW i dt W V W i , i V i i - Time constant - Represents the effect of temperature 23 The Hodgkin & Huxley Framework The current through each channel has the form: I j g j j V , W 1 , , W N (V Vj) g j - Maximal conductance (when all channels are open) j - Fraction of open channels (can depend on several W variables). 24 The Temperature Parameter Φ • Allows for taking into account different temperatures. • Increasing the temperature accelerates the kinetics of the underlying processes. • However, increasing the temperature does not necessarily increase the excitability. Both increasing and decreasing the temperature can cause the neuron to stop firing. • A phenomenological model for Φ is: 3 Temp 6 . 3 / 10 25 Hodgkin & Huxley Model dm/dt m (V)-m /τ m dh/dt h (V)-h /τ h dn/dt n (V)-n /τ n 26 Ionic Conductances During an Action Potential 27 Repetitive Firing in Hodgkin–Huxley Model A: Voltage time courses in response to a step of constant depolarizing current. from bottom to top: Iapp= 5, 15, 50, 100, 200 in μamp/cm2). Scale bar is 10 msec. B: f-I curves for temperatures of 6.3,18.5, 26◦C, as marked. Dotted curves show frequency of the unstable periodic orbits. 28 Fast-Slow Dissection of the Action Potential • n and h are slow compared to m and V. • Based on this observation, the system can be dissected into two time-scales. • This simplifies the analysis. • For details see: Borisyuk A & Rinzel J. Understanding neuronal dynamics by geometrical dissection of minimal models. In, Chow et al, eds: Models and Methods in Neurophysics (Les Houches Summer School 2003), Elsevier, 2005: 19-72. 29 Correlation between n and h • During the action potential the variables n and h vary together. • Using this correlation one can construct a reduced model. • The first to observe this was Fitzhugh. 30 Simplified Versions of the HH Model • Models that generate action potentials can be constructed with fewer dynamic variables. • These models are more amenable for analysis and are useful for learning the basic principles of neuronal excitability. • We will focus on the model developed by Morris and Lecar. 31 The Morris-Lecar Model (1981) • Developed for studying the barnacle muscle. • Model equations: C dV dt dw dt I ion V , w I app ( t ) w V w w V I ion V , w g Ca m (V )( V V Ca ) g K w (V V K ) g L (V V L ) 32 Morris-Lecar Model • The model contains K and Ca currents. • The variable w represents the fraction of open K channels. • The Ca conductance is assumed to behave in an instantaneous manner. m (V ) 0 . 5 1 tanh V V1 / V 2 w (V ) 0 . 51 tanh V V 3 / V 4 w V 1 / cosh V V 3 / 2V 4 33 Morris-Lecar Model • A set of parameters for example: V1 1 . 2 g Ca 4 V Ca 120 V 2 18 gK 8 V K 84 V3 2 gL 2 V L 60 μF C 20 cm 2 0 . 04 V 4 30 34 Morris-Lecar Model • Voltage dependence of the various parameters (at long times): 35 Conductance-Based Models of Cortical Neurons 36 Conductance-Based Models of Cortical Neurons • Cortical neurons behave differently than the squid axon that Hodgkin and Huxley investigated. • Over the years, people developed several variations of the HH model that are more appropriate for describing cortical neurons. • We will now see an example of a simple model which will later be useful in network simulations. • The model was developed by playing with the parameters such that its f-I curve is similar to that of cortical neurons. 37 Frequency-Current Responses of Cortical Neurons Excitatory Neuron: Ahmed et. al., Cerebral Cortex 8, 462-476, 1998 Inhibitory Neurons: Azouz et. al., Cerebral Cortex 7, 534-545, 1997 38 Frequency-Current Responses of Cortical Neurons The experimental findings show what f-I curves of cortical neurons are: • Continuous – starting from zero frequency. • Semi-Linear – above the threshold current the curve is linear on a wide range. How can we reconstruct this behavior in a model? 39 An HH Neuron with a Linear f-I Curve Shriki et al., Neural Computation 15, 1809–1841 (2003) 40 Linearization of the f-I Curve • We start with an HH neuron that has a continuous f-I curve (type I, saddle-node bifurcation). • The linearization is made possible by the addition of a certain K-current called A-current. • The curve becomes linear only when the time constant of the A-current is slow enough (~20 msec). • There are other mechanisms for linearizing f-I curves. 41 Model Equations: Shriki et al., Neural Computation 15, 1809–1841 (2003) 42