1. Basic Models in Theoretical Neuroscience Oren Shriki 2010 Integrate and Fire and Conductance Based Neurons 1

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3. 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

4. The Neuron as an Electric Circuit 4

5. Intracellular Recording 5

6. 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

7. Ion Channels and Ion Pumps 7

8. The Neuron as an Electric Circuit Differences in ionic concentrations  Battery Cell membrane  Capacitor Ionic channels  Resistors 8

9. The Neuron as an Electric Circuit Extracellular Intracellular 9

10. RC circuits R – Resistance (in Ohms) C – Capacitance (in Farads) I R C Current source 10

11. RC circuits I R C The dynamical equation is: 11

12. RC circuits Defining: We obtain: The general solution is: 12

13. RC circuit Response to a step current: 13

14. RC circuit Response to a step current: 14

15. The Integrate-and-Fire Neuron R – Membrane Resistance (1/conductance) C – Membrane Capacitance (in Farads) I R C inside outside ELEL Threshold mechanism 15

16. Integrate-and-Fire Neuron If we define: The dynamical equation will be: To simplify, we define: Thus: 16

17. 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, τ R. 17

18. Integrate-and-Fire Neuron Response to a step current: IR<θ: t V 18

19. Integrate-and-Fire Neuron Response to a step current: IR>θ: V t T τRτR τRτR τRτR 19

20. Integrate-and-Fire Neuron Finding the firing rate as a function of the applied current: 20

21. Integrate-and-Fire Neuron f I 21

22. The Hodgkin-Huxley Equations 22

23. The Hodgkin & Huxley Framework Each gating variable obeys the following dynamics: - Represents the effect of temperature - Time constant 23

24. The Hodgkin & Huxley Framework The current through each channel has the form: - Maximal conductance (when all channels are open) - Fraction of open channels (can depend on several W variables). 24

25. 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: 25

26. Hodgkin & Huxley Model 26

27. Ionic Conductances During an Action Potential 27

28. 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

29. 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

30. 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

31. 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

32. The Morris-Lecar Model (1981) Developed for studying the barnacle muscle. Model equations: 32

33. 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. 33

34. Morris-Lecar Model A set of parameters for example: 34

35. Morris-Lecar Model Voltage dependence of the various parameters (at long times): 35

36. Conductance-Based Models of Cortical Neurons 36

37. 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

38. 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

39. 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

40. An HH Neuron with a Linear f-I Curve Shriki et al., Neural Computation 15, 1809–1841 (2003) 40

41. 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

42. Model Equations: Shriki et al., Neural Computation 15, 1809–1841 (2003) 42