synapses and multi compartmental models computational neuroscience 03 lecture 2

Synapses and Multi Compartmental Models

Computational Neuroscience 03

Lecture 2

Synapses:

The synapse is remarkably complex and involves many simultaneousprocesses such as the production and degredation of neurotransmitter.The neurotransmitters directly (A) or indirectly (B) binds to a synaptic channel and activates it.

Synaptic conductances:

Synaptic transmission begins when an action potential invades the presynaptic terminal and activates voltage dependent Ca2+ channels.

This causes transmitter molecules to enter the cleft and bind to receptors on the postsynaptic neuron.

As a result ion channels open, which modifies the conductance of the postsynaptic neuron

Pgg ss Synaptic conductance:P: open channel probability

relsPPP

Prel: probability of transmitter releasePs: probability that postsyn. channel opensBoth stochastic processes

SS PPS

S

1

Postsynaptic conductance:

SSSSS PP

dt

dP )1(

.constS closing rate of the channel

S opening rate dependent on transmitter conc

S

Spike

T

t0t

:0t )0(SP

SS ignore during the opening process

S

tSS

SePtP )1)0((1)( for

for)()()( TtSS

SeTPtP

Tt 0

Tt

if there is no synaptic release immediately before the releaseat t=0 0)0( SP

TS

SeTPP 1)(max

using a simple manipulation we can write in the general case

))0(1()0()( max SSS PPPTP Update rule after spikes

Fast synapse:

For a fast synapse the rise of the conductance following apresynaptic action potential can be approximated asinstantaneous.For a single presynaptic action potential occurring at t=0 wecan write

S

t

S ePP

max SS

1with

A sequence of action potentials at arbitrary times can be modeledwith an exponential decay

SS

S Pdt

dP

and by updating the probability after eachaction potential with: )1(max SSS PPPP

Slow synapse (e.g. GABAA and NMDA):

21

max)( tt

S eeBPtP

For an isolated presynaptic action potential occurring at t=0 wecan use the same model or a difference of two exponentials

21 1/

1

2

/

1

2

21

riserise

B

21

21

rise

or the alpha function

S

t

SS e

tPP

1

maxwith a peak value at St

B is a normalizationfactor and ensures thatthe peak value is equalto Pmax

Examples of time-dependent open probabilities:

Single exp. decay Diff of two exp.

exponential decay

Instantaneous rise

))(()( NMDASNMDANMDANMDA EVtPVGgi

SSSSS PP

dt

dP )1(

NMDA:

Slow (20ms rise)

Physiological correlate of the Hebblearning rule since both, the presynapticand postsynaptic cell have to be active.

The voltage dependence is mediated bymagnesium ions which normally blockNDMA receptors. The postsynaptic cellMust be sufficiently depolarized to knockout the blocking ions. Dependence of the NMDA conductance

on the membrane potential V and theextracellular Mg2+ concentration.

Probability of transmitter release and short-term plasticity:

Depression (D) and facilitation (F)of excitory intercortical synapses

relrel

P PPdt

dP 0

with 0P the release probability after a long period of silence

)1( relFrelrel PfPP

relDrel PfP Threshold

Steady-state release probability for a presynaptic Poisson spike-train:

relP average steady state release probability

)1( relFrelrel PfPP The facilitationafter each spike is cancelled out by the average exponentialdecrement between presynaptic spikes.

Consider two action potentials separated by an intervaland the release probability at the time of the first spike is

relP

relFrelrel PfPP 1

Immediately after the spike the release probabilityis set to

By the time of the second spike it is decayed to pePPfPPP relFrelrel

00 1

Since we are interested in the average release probability, wehave to determine the average exponential decay with aninterspike interval

000 )(1

dtpePPfPPPr

p

re

isirelFrelrel

Probability density of aPoisson spike train withinterspike intervall .

p

p

rr

r

rderder p

p

p

10

)1

(

0

)1

(

p

prelFrelrel r

rPPfPPP

11 00

pF

pFrel rf

rfPP

1

0

Steady-state release probability for a presynaptic Poisson spike-train:

pDrel rf

PP

)1(10

Facilitating synapse Depressing synapse

relPr : Synaptic transmission

Transmission for a depressing synapse:

Due to the 1/r release probability at high rates, the synaptic transmission becomesindependent of the firing rate. Thus, depressing synapses do not convey the valueof high presynaptic firing rates. They emphasize changes in the firing rate.

pDrel rf

rPPr

)1(10

Prior to a change:

r

r

rf

PrrPr

pDrel

1)1(1

)(' 0

After a change: for high

rates r

))(( AMPASAMPAAMPA EVtPgi

SSSSS PP

dt

dP )1(

AMPA:

fast

Glutamate activates two different kinds of receptors: AMPA and NMDA.

Both receptors lead to an excitation of the membrane.

Examples of some synapses:

GABA (aminobutyric acid) is the principal inhibitoryneurotransmitter.There are two main receptors for GABA, GABAA and GABAB.

GABAA

GABAA is responsible for fast inhibtion and require onlybrief stimuli to produce a response.

))((AAA GABASGABAGABA EVtPgi

SSSSS PP

dt

dP )1(

GABAB

GABAB is a much more complex receptor. It involves so-calledsecond messengers. GABAB responses occur when the GABAbinds to another compound (G-potein) which in turn binds to aPotassim channel and opens it up. It takes 4 activated G-proteinsto open the channel.

)(4

4

K

dS

SGABAGABA EV

KP

Pgi

BB

SrS PKPK

dt

dP43

rrrSr PP

dt

dP )1(

Gap junctions are not chemical synapses but electrical in nature.The produce a current proportional to the difference betweenpre-and postsynaptic potential. No transmitter or action potentialis involved. Many non-neural cells, e.g. muscle, glia, are coupledin this manner.

)( prepostCgap VVgi

Synaptic in integrate and fire

emsssmLm IREVPgrVEdt

dVc )(

Can also add synapses onto an integrate and fire using:

Ps is probability of firingand changes whenever the presynaptic neuron fires using one of the schemes mentioned previously. For simplicity, assume Prel is 1

Used to look at eg dynamics of large numbers of inputs and effects of inhibitory/excitatory synapses

Have excitatory or inhibitory synapses depending on whether ES is above/below membrane potential

interestingly inhibitory synapse produces more synchronous firing

Cable model

Have been assuming no spatial variation in membrane potential: Not true especially for neurons with long narrow processesAttenuation and degradation of signal is most severe when current passes down the long cable-like dendritic or axonal branches=> Cable theory: assumption is that cables are radially homogeneousLongtitudinal resistance over cable of length x:

RL = rL x/(a2)

Where rL is intracellular reistivity and a is radius

Simultaneous recordings from different parts of neurons

From Ohm’s law, we get a pde relating voltage difference to current im (see abbott and Dayan, pp204-207)

im generally complex and must be integrated numerically. However for linear current, no synaptic current and infinite cable can solve analytically to get

Where electrotonic length

L

m

r

ar

2

And voltage drops by a factor of exp(-n) at a distance of

This model made more realistic by Rall: equivalent cable model

Multi Compartment models• Dendrites only locally uniform

• Make different compartments with different properties

• Each compartment reduces to the cable model

• Need some way of them linking at the ends:

)()( 11,11,

VVgVVg

A

Ii

dt

dVc e

mm

Where compartments are indexed by

More compartments means better approximation

However, splitting axons into locally unifrom sections OK since we have to do this in numerical integration

The simplest method of solving ODE's is Euler's method:

)()( tzmhhtzh

Rule of the thumb:10min

h

Piecewise approximation:

)()()(

tRIVtVdt

tdVerestm

m

)()(

tVVdt

tdVm

m

For constant I:

0

)()( 0

tt

eVtVVtV

h

eVtVVhtV

)()( Vupdate

The value of the conductance between compartments can be calculated from Ohm’s law. If compartments have equal length L and radius a

If not:

Can solve the equations and so see how APs propagate along axons.

)( 2'

2'

2'

',

aLaLLr

aag

L

2', 2 Lr

ag

L

AP PropagationAP propagates since point 0 is depolarised to V0 it causes pt 1 to depolarise to V1. Before reaching V1 however it crosses threshold causing spike, which causes pt 2 which had been moving towards V2 to move towrds V1 etc

APs can move either way, but don’t go backwards because of refractory period. Speed of propagation can be shown to be proportional to sqrt(radius) for unmyelinated axons. However, for myelinated axons, at the optimal thickness of myelin inner=0.6outer which is actual thickness of axons (thickness effects membrane capacitance etc) with respect to speed, speed is proportional to radius