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 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

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. 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.

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

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

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 V g (V, I )=0

Kirchhoff’s Voltage Law The directed sum of electrical potential differences around a circuit loop is 0. To apply this law: 1) Choose the orientation of the loop. 2) Sum the voltages to zero (“+” if its current is of the same direction as the orientation and “-” if current is opposite the orientation).

Kirchhoff’s Current Law The directed sum of the currents flowing into a point is zero. To apply this law: 1) Choose the directions of the current branches. 2) Sum the currents to zero (“+” if a current points toward the point and “-” if it points away from the point).

Example  By Kirchhoff’s Voltage Law  with Device Relationships  and substitution to get or

Circuit Models of Neurons I = F(V)

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

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

-I (t)

Morris, C. and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophysical J., 35(1981), pp 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 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 Izhikevich, E.M Neural excitability, spiking, and bursting, Int. J. Bif. & Chaos, 10(2000), pp (also see his article in SIAM Review) (Non-circuit) Models for Excitable Membranes

Our Circuit Models

 By Ion Pump Characteristics  with substitution and assumption  to get Equations for Ion Pumps

Dynamics of Ion Pump as Battery Charger

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

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 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 Equivalent IV-Characteristics --- for serial potassium channels 0

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

C = Rinzel & Wang (1997) Bursting Spikes

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. 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.

Metastability and Plasticity

C = C = 0.5 Neural Chaos C = 0.5 = 0.05  = 0.18  = I in = 0 g K = d K = i 1 = 0.14 i 2 = 0.52 E K = g Na = 1 d Na = v 1 = v 2 = E Na = 0.6

Myelinated Axon with Multiple Nodes Inside the cell Outside the cell

Signal Transduction along Axons Neuroscience: 3ed

Circuit Equations of Individual Node

Coupled Equations for Neighboring Nodes Couple the nodes by adding a linear resistor between them

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:

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

C=.7 pF

Transmission Speed C=.01 pFC=.1 pF

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. 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,  Metastability and Plasticity of Conceptual Circuit Models of Neurons, Journal of Integrative Neuroscience, References:  A Conceptual Circuit Model of Neuron, Journal of Integrative Neuroscience,  Metastability and Plasticity of Conceptual Circuit Models of Neurons, Journal of Integrative Neuroscience, 2010.