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NENS220 Computational methods in Neuroscience

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NENS220 Computational methods in Neuroscience

John Huguenard and Terry Sanger

- Overview of computational methods
- Mathematical techniques for creating models of neural behavior - the tools of computational methods

- The ultimate purpose is to relate different levels (scales) of neural behavior
- e.g.: how do properties of ion channels determine the spiking behavior in response to synaptic input?
- e.g.: what is the relationship between spike activity in a population of M1 neurons and movement of the arm?

- This is essentially an overview of some (but not all) of the general methods
- Intended for graduate students in neuroscience
- In order to learn how this is done, you will have to practice
- Necessarily involves knowledge of statistics, mathematics, and some computer programming (matlab, NEURON)

- Probability theory
- Information theory
- Matrix algebra
- Correlation integrals
- Fourier analysis
- Matlab programming
- Membrane potentials
- Cable theory

- We will do much of this as we go.
- Additional help in TA sessions
- You may need to do extra reading

- I: Neurons
- How information is processed at the level of synapses, membranes, and dendrites
- Relationship between inputs, membrane potentials, and spike generation

- II: Spikes:
- What information is carried in single spikes, temporal sequences of spikes, and spikes over populations
- How learning results in changes in spike patterns

- Theoretical Neuroscience, Peter Dayan and Larry Abbott, (MIT Press: Cambridge MA), 2001.
- Available from Amazon.com and the Stanford bookstore, about $45
- Other useful references:
- Neural Engineering, Andersen and Elliasmith
- Spikes, Bialek
- Computational Neuroscience, Churchland and Sejnowski
- Handbook of computational neuroscience, Arbib
- Foundations of Cellular Neurophysiology, Johnston and Wu

- You must have access to a workstation with matlab/NEURON.
- Matlab available on cluster computers (firebirds, etc)
- NEURON available for multiple platforms via free download
- We can set up accounts on linux machines with NEURON installed.

- Tuesdays and Thursdays, 3:15-5:00pm. Room H3150.
- Tuesdays will be lectures
- Lecture will usually follow the text chapters; you may want to read these in advance
- A paper will be assigned, to be read before Thursday (first paper assigned next Tuesday)

- Thursdays will be discussions of the assigned paper and the lecture, led by the TA. THESE ARE REQUIRED.

- 4-6 problems sets during quarter. They will be assigned on Tuesday and due the Thursday of the following week (9 days later)
- Will usually require simulation of some component of the paper being discussed.
- Will require use of matlab/NEURON. You should submit the program output and source code with detailed comments
- Should require 1-3 hours, depending on how good you are with matlab/NEURON

- 70% weekly assignments
- Based on output plots, code, and comments

- 30% Class participation
- Based on contributions to discussion groups

Modeling of realistic neurons and networks

John Huguenard

Neuroelectronics, Part I

John Huguenard

- An “external signal” x(t) is something that the experimenter controls (either a sensory stimulus or a learned motor task)
- We observe spikes that are the result of a transformation of the external signal

External World

Sensors

Spike Generator

x(t)

spikes

- Neurons receive synaptic input
- Neurons produce output
- The currency of neuronal communication is spikes (action potentials)
- Spike generation is in many cases a nonlinear function of synaptic input

External World

Sensors

Spike Generator

x(t)

spikes

- STDP, dendritic back propagation, dendritic signaling
- Resonance
- Oscillations
- Synchronization
- Gain control
- Persistent activity
- Phase precession
- Coincidence detection vs. integration

Pyramidal Neurons in Layer V

thy1-YFP mouse

Feng et al., (2000) Neuron 28:41

200 µm

Canonical Microcircuits

Recurrent excitatory connections are prominent.

Function: Amplification of signals for enhanced feature detection.

Rodney Douglas & Kevan Martin

I

II

III

IV

V

VI

Modified From: Karube et al., (2004) J Neurosci 24:2853-65

Mainen & Sejnowski, 1996

Electrical properties of neurons

Dominated by membrane capacitance

Neurons are integrators

whose time constant is dynamically variable

Spike output depends on voltage-dependent gating of ion channels

Passive properties of neurons

Semipermeable lipid bilayer membrane with high [K+]i maintained by electrogenic pump (ATPase)

Equivalent radius ~ 25 mm,

Surface Area ~ 8000 mm2=.008 mm2=8e-5cm2

_

+

- Ability to store charge
- Charge required to create potential difference between two conductors
- A 1 Farad capacitor will store 1 Coulomb/Volt

Hille, 2001

_

+

- Capacitance is a function of
- Surface area (A)
- Dielectric constant (e)
- Distance between plates (d)

- For membranes specific capacitance 1 mF/cm2
- is for the most part invariant
- for a 0.8e-5cm2 cell ~ 80 pF

Hille, 2001

Nernst Potentials

EK = -75 mV

ENa = +50 mV

ECl = -60 mV

ECa = +100 mV

Nernst equation:

- [K+]i 130 mM
- [K+]o 3 mM
- EK –100 mV
- q=CV
- = 80 pF*100mV
- = 8pC
- = 50e6 K+ ions
- Total K+I = 5e12 ions
- Fraction uncompensated = 0.001%
- Will vary with surface to volume ratio

- Ion selective pores
- Ohm’s law E = IR
- 1V is the potential difference produced by 1A passing through 1 Ohm

- Conductance is reciprocal of resistance,1 Siemen = 1 Ohm-1
- Resistance is dependent on length of conductive path, cross sectional area, and resistivity of the media
- Ion channels have conductances in the 2-250 pS range, but may open only briefly

- Ohm’s law E = IR

- “Leak” channels are open at rest and determine the input resistance
- i.e. the impedance to extrinsic current injection

- Specific input resistance for neurons is in the range of 1 MW mm2 or 1 mS/mm2
- 50,000 20 pS leak channels/mm2 = 1 channel / 20 mm2

- Our “typical” cell of 0.008 mm2 would have an input resistance of 125 MW, or input conductance of 8 nS (equivalent to ~400 open leak channels).

Parallel conductance model

- Characterized by an open channel I/V that is linear
- I = (Vrev- Vm)/R
- I = (Vrev- Vm) * g

g = slope

Vrev

- Goldman-Hodgkin-Katz (GHK) theory
- Ions pass independently
- Electrical field within membrane is constant
- sometimes known as the constant field equations

- GHK current equation (flux in two directions)
- GHK Voltage equation

- GHK current equation

[S]i>[S]o

Better description than

ohmic for some channels

e.g. Ca2+, K+

- Channel opening is a function of transmembrane voltage

- Stable systems have positive slope to I/V curve. E.g., neurons with only leak currents.
- Voltage dependent conductances can lead to regions of negative slope conductance with two stable states.

- gK:gNa =3:1, both linear

- gK:gNa =3:5

- Synaptic input
- Transient inputs from other sources
- i.e. sensors or lower level neurons

- Spike output
- Generation of action potentials, which will then propagate the signal to the neurons at the next level, again via synapses

- Excitatory (ES > Em)
- Inhibitory (Es < Em)
- shunting

- Rapid increase in gs followed by exponential decay (tD = 1 - 100 ms)
- Approximated by alpha function
- Or sum of exponentials (more realistic)

- Nonlinear, “all or none” response
- Based on avalanche type reaction
- Characterized by
- Threshold
- High conductance reset
- Refractory period

- Can be simulated by integrate and fire synthetic neuron