NENS220 Computational methods in Neuroscience

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NENS220 Computational methods in Neuroscience. John Huguenard and Terry Sanger. Goals of the course. Overview of computational methods Mathematical techniques for creating models of neural behavior - the tools of computational methods. Computational Modeling.

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### NENS220 Computational methods in Neuroscience

John Huguenard and Terry Sanger

Goals of the course
• Overview of computational methods
• Mathematical techniques for creating models of neural behavior - the tools of computational methods
Computational Modeling
• 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?
Scope of the course
• 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)
Background material
• Probability theory
• Information theory
• Matrix algebra
• Correlation integrals
• Fourier analysis
• Matlab programming
• Membrane potentials
• Cable theory
Background review
• We will do much of this as we go.
• Additional help in TA sessions
• You may need to do extra reading
Two major areas
• 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
Textbook
• 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
• Matlab available on cluster computers (firebirds, etc)
• We can set up accounts on linux machines with NEURON installed.
Class structure
• Tuesdays and Thursdays, 3:15-5:00pm. Room H3150.
• Tuesdays will be lectures
• 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.
Homework assignments
• 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
Part I

Modeling of realistic neurons and networks

John Huguenard

### Neuroelectronics, Part I

John Huguenard

The big picture, a la Terry Sanger
• 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

The big picture, a la John Huguenard
• 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

Why it is important to consider neuronal properties..
• 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

Inhibitory interneuron diversity

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

### 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)

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

_

+

Electrical capacitance
• 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 of cell membranes
• 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

Resting potential, single permeant ion

Nernst Potentials

EK = -75 mV

ENa = +50 mV

ECl = -60 mV

ECa = +100 mV

Nernst equation:

Uncompensated charge
• [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
Membrane Resistance
• 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
Input Resistance
• “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).
Ohmic channels
• Characterized by an open channel I/V that is linear
• I = (Vrev- Vm)/R
• I = (Vrev- Vm) * g

g = slope

Vrev

Non-ohmic channels
• 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
Nonlinear driving force
• GHK current equation

[S]i>[S]o

Better description than

ohmic for some channels

e.g. Ca2+, K+

Voltage dependent conductances
• Channel opening is a function of transmembrane voltage
Latching, up- and down-states
• 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.
Two types of channels
• gK:gNa =3:1, both linear
Signalling
• 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
Chemical 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)
Spike generation
• 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