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Computing in carbon

Explore basic elements of neuroelectronics, from membranes and ion channels to wiring neurons and synapses. Learn about conductance-based models, equilibrium potentials, excitability of neurons, and advanced spike response models.

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Computing in carbon

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  1. Computing in carbon Basic elements of neuroelectronics -- membranes -- ion channels -- wiring Elementary neuron models -- conductance based -- modelers’ alternatives Wiring neurons together -- synapses -- short term plasticity

  2. Closeup of a patch on the surface of a neuron

  3. An electrophysiology experiment = 1 mF/cm2 Ion channels create opportunities for charge to flow Potential difference is maintained by ion pumps

  4. Movement of ions through the ion channels Energetics: qV ~ kBT V ~ 25mV

  5. until opposed by electrostatic forces Ions move down their concentration gradient The equilibrium potential K+ Na+, Ca2+ Nernst:

  6. ENa V 0 more polarized Vrest EK Different ion channels have associated conductances. A given conductance tends to move the membrane potential toward the equilibrium potential for that ion depolarizing depolarizing hyperpolarizing shunting ENa ~ 50mV ECa ~ 150mV EK ~ -80mV ECl ~ -60mV V > E  positive current will flow outward V < E  positive current will flow inward

  7. The neuron is an excitable system

  8. Excitability is due to the properties of ion channels • Voltage dependent • transmitter dependent (synaptic) • Ca dependent

  9. K channel: open probability increases when depolarized The ion channel is a complex machine n describes a subunit n is open probability 1 – n is closed probability Transitions between states occur at voltage dependent rates C  O PK ~ n4 O  C Persistent conductance

  10. Transient conductances Gate acts as in previous case Additional gate can block channel when open PNa ~ m3h m is activation variable h is inactivation variable m and h have opposite voltage dependences: depolarization increases m, activation hyperpolarization increases h, deinactivation

  11. First order rate equations We can rewrite: where

  12. A microscopic stochastic model for ion channel function approach to macroscopic description

  13. Transient conductances Different from the continuous model: interdependence between inactivation and activation transitions to inactivation state 5 can occur only from 2,3 and 4 k1, k2, k3 are constant, not voltage dependent

  14. - Putting it back together and Kirchhoff’s law Ohm’s law: Capacitative current Ionic currents Externally applied current

  15. The Hodgkin-Huxley equation

  16. Anatomy of a spike

  17. The integrate-and-fire model Like a passive membrane: but with the additional rule that when V  VT, a spike is fired and V  Vreset. EL is the resting potential of the “cell”.

  18. The spike response modelGerstner and Kistler Kernel f for subthreshold response  replaces leaky integrator Kernel for spikes  replaces “line” • determine f from the linearized HH equations • fit a threshold • paste in the spike shape and AHP

  19. An advanced spike response modelKeat, Reinagel and Meister • AHP assumed to be exponential recovery, A exp(-t/t) • need to fit all parameters

  20. The generalized linear model Paninski, Pillow, Simoncelli • general definitions for k and h • robust maximum likelihood fitting procedure

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