1. Synapses and Multi Compartmental Models
Computational Neuroscience 03
Lecture 2

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

3. 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
P: open channel probability
Synaptic conductance:
Prel: probability of transmitter release
Ps: probability that postsyn. channel opens
Both stochastic processes

4. Postsynaptic conductance:
closing rate of the channel
opening rate dependent on transmitter conc
ignore during the
opening process
Spike

5. for
for
if there is no synaptic release immediately before the release
at t=0
using a simple manipulation we can write in the general case
Update rule after spikes

6. Fast synapse:
For a fast synapse the rise of the conductance following a
presynaptic action potential can be approximated as
instantaneous.
For a single presynaptic action potential occurring at t=0 we
can write
with
A sequence of action potentials at arbitrary times can be modeled
with an exponential decay
and by updating the probability after each
action potential with:

7. Slow synapse (e.g. GABAA and NMDA):
For an isolated presynaptic action potential occurring at t=0 we
can use the same model or a difference of two exponentials
B is a normalization
factor and ensures that
the peak value is equal
to Pmax
or the alpha function
with a peak value at

8. Examples of time-dependent open probabilities:
Instantaneous rise
Single exp. decay
Diff of two exp.
exponential decay

9. NMDA: Slow (20ms rise) Physiological correlate of the Hebb
learning rule since both, the presynaptic
and postsynaptic cell have to be active.
The voltage dependence is mediated by
magnesium ions which normally block
NDMA receptors. The postsynaptic cell
Must be sufficiently depolarized to knock
out the blocking ions.
Dependence of the NMDA conductance
on the membrane potential V and the
extracellular Mg2+ concentration.

10. Probability of transmitter release and short-term plasticity:
Depression (D) and facilitation (F)
of excitory intercortical synapses
Threshold
with
the release probability after a long period of silence

11. Steady-state release probability for a presynaptic Poisson spike-train:
average steady state release probability
The facilitation
after each spike is cancelled out by the average exponential
decrement between presynaptic spikes.
Consider two action potentials separated by an interval
and the release probability at the time of the first spike is
Immediately after the spike the release probability
is set to
By the time of the second spike it is decayed to

12. Since we are interested in the average release probability, we
have to determine the average exponential decay with an
interspike interval
Probability density of a
Poisson spike train with
interspike intervall t.

13. Steady-state release probability for a presynaptic Poisson spike-train:
Facilitating synapse
Depressing synapse
: Synaptic transmission

14. Transmission for a depressing synapse:
Due to the 1/r release probability at high rates, the synaptic transmission becomes
independent of the firing rate. Thus, depressing synapses do not convey the value
of high presynaptic firing rates. They emphasize changes in the firing rate.
Prior to a change:
for high
rates r
After a change:

15. Examples of some synapses:
Glutamate activates two different kinds of receptors:
AMPA and NMDA.
Both receptors lead to an excitation of the membrane.
AMPA:
fast

16. GABA (g-aminobutyric acid) is the principal inhibitory
neurotransmitter.
There are two main receptors for GABA, GABAA and GABAB.
GABAA
GABAA is responsible for fast inhibtion and require only
brief stimuli to produce a response.

17. GABAB
GABAB is a much more complex receptor. It involves so-called
second messengers. GABAB responses occur when the GABA
binds to another compound (G-potein) which in turn binds to a
Potassim channel and opens it up. It takes 4 activated G-proteins
to open the channel.

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

19. Synaptic in integrate and fire
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

20. 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

21. Cable model Simultaneous recordings from different parts of neurons
Have been assuming no spatial variation in membrane potential: Not true especially for neurons with long narrow processes
Attenuation 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 homogeneous
Longtitudinal resistance over cable of length Dx:
RL = rL Dx/(pa2)
Where rL is intracellular reistivity and a is radius

22. From Ohm’s law, we get a pde relating voltage difference to current im (see abbott and Dayan, pp )
im generally complex and must be integrated numerically. However for linear current, no synaptic current and infinite cable can solve analytically to get
And voltage drops by a factor of exp(-n) at a distance of l
Where electrotonic length

23. This model made more realistic by Rall: equivalent cable model

24. 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:
Where compartments are indexed by m

25. More compartments means better approximation

26. 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:
Rule of the thumb:

27. Piecewise approximation:
For constant I:
update

28. 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.

29. AP Propagation
AP 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

30. 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