Exploring Neural Circuit Dynamics: Models and Examples

1. Neural Circuits & Dynamics Bo Deng University of Nebraska-Lincoln • Topics: • Circuit Basics • Circuit Models of Neurons • --- FitzHuge-Nagumo Equations • --- Hodgkin-Huxley Model • --- Our Models • Examples of Dynamics • --- Bursting Spikes • --- Metastability and Plasticity • --- Chaos • --- Signal Transduction Joint work with undergraduate and graduate students: Suzan Coons, Noah Weis, Adrienne Amador, Tyler Takeshita, Brittney Hinds, and Kaelly Simpson

2. Circuit Basics • Q = Q(t) denotes the net positive charge at a point of a circuit. • I = dQ(t)/dt defines the current through a point. • V = V(t)denotes the voltage across the point. • Analysis Convention: When discussing current, we first assign • a reference direction for the current I of each device. Then we have: • I > 0 implies Q flows in the reference direction. • I < 0 implies Q flows opposite the reference direction.

3. Capacitors Review of Elementary Components • A capacitor is a device that stores energy in an electric potential field. Q

4. Inductors • An inductor is a device that stores (kinetic) energy in a magnetic field. dI/dt

5. Resistors • A resistor is an energy converting device. • Two Types: • Linear • Obeying Ohm’s Law: V=RI, where R is resistance. • Equivalently, I=GV with G = 1/R the conductance. • Variable • Having the IV – characteristic constrained by an equation g (V, I )=0. I g (V, I )=0 V

6. Kirchhoff’s Voltage Law • The directed sum of electrical potential differences around a circuit loop is 0. • To apply this law: • Choose the orientation of the loop. • Sum the voltages to zero (“+” if its current is of the same direction as the orientation and “-” if current is opposite the orientation).

7. Kirchhoff’s Current Law • The directed sum of the currents flowing into a point is zero. • To apply this law: • Choose the directions of the current branches. • Sum the currents to zero (“+” if a current points toward the point and “-” if it points away from the point).

8. Example • By Kirchhoff’s Voltage Law • with Device Relationships • and substitution to get • or

9. Circuit Models of Neurons I = F(V)

10. Excitable Membranes • Kandel, E.R., J.H. Schwartz, and T.M. Jessell • Principles of Neural Science, 3rd ed., Elsevier, 1991. • Zigmond, M.J., F.E. Bloom, S.C. Landis, J.L. Roberts, and L.R. Squire • Fundamental Neuroscience, Academic Press, 1999. Neuroscience: 3ed

11. Kirchhoff’s Current Law Hodgkin-Huxley Model - I (t)

12. -I (t)

13. (Non-circuit) Models for Excitable Membranes • Morris, C. and H. Lecar, • Voltage oscillations in the barnacle giant muscle fiber, • Biophysical J., 35(1981), pp.193--213. • Hindmarsh, J.L. and R.M. Rose, • A model of neuronal bursting using three coupled first order differential • equations, • Proc. R. Soc. Lond. B. 221(1984), pp.87--102. • Chay, T.R., Y.S. Fan, and Y.S. Lee • Bursting, spiking, chaos, fractals, and universality in biological • rhythms, Int. J. Bif. & Chaos, 5(1995), pp.595--635. • Izhikevich, E.M • Neural excitability, spiking, and bursting, • Int. J. Bif. & Chaos, 10(2000), pp.1171--1266. • (also see his article in SIAM Review)

14. Our Circuit Models

17. Equations for Ion Pumps • By Ion Pump Characteristics • with substitution and assumption • to get

18. Dynamics of Ion Pump as Battery Charger

19. Equivalent IV-Characteristics --- for parallel sodium channels Passive sodium current can be explicitly expressed as

20. Equivalent IV-Characteristics --- for serial potassium channels Passive potassium current can be implicitly expressed as A standard circuit technique to represent the hysteresis is to turn it into a singularly perturbed equation 0

22. Examples of Dynamics --- Bursting Spikes --- Metastability & Plasticity --- Chaotic Shilnikov Attractor --- Signal Transduction Geometric Method of Singular Perturbation • Small Parameters: • 0 < e<< 1 with ideal • hysteresis at e = 0 • both C and l have • independent time scales

23. Rinzel & Wang (1997) Bursting Spikes C = 0.005

24. Metastability and Plasticity • Terminology: • A transient state which behaves like a steady state is • referred to as metastable. • A system which can switch from one metastable state • to another metastable state is referred to as plastic.

25. Metastability and Plasticity

26. Neural Chaos gNa = 1 dNa = - 1.22 v1 = - 0.8 v2 = - 0.1 ENa = 0.6 • C = 0.5 • = 0.05 • g = 0.18 • = 0.0005 • Iin = 0 C = 0.005 gK = 0.1515 dK = -0.1382 i1 = 0.14 i2 = 0.52 EK = - 0.7 C = 0.5

28. Myelinated Axon with Multiple Nodes Inside the cell Outside the cell

29. Signal Transduction along Axons Neuroscience: 3ed

30. Neuroscience: 3ed

31. Neuroscience: 3ed

32. Circuit Equations of Individual Node

33. Coupled Equations for Neighboring Nodes • Couple the nodes by adding a linear resistor between them Current between the nodes

34. The General Case for N Nodes • This is the general equation for the nth node • In and out currents are derived in a similar manner:

35. C=.1 pF C=.7 pF (x10 pF)

36. C=.7 pF

37. Transmission Speed C=.1 pF C=.01 pF

39. Closing Remarks: • The circuit models can be further improved by dropping the • serial connectivity of the passive electrical and • diffusive currents. • Existence of chaotic attractors can be rigorously proved, • including junction-fold, Shilnikov, and canard attractors. • Can be fitted to experimental data. • Can be used to form neural networks. • References: • A Conceptual Circuit Model of Neuron, Journal of Integrative Neuroscience, 2009. • Metastability and Plasticity of Conceptual Circuit Models of Neurons, Journal of Integrative Neuroscience, 2010.