Download Exercise 15 : Cable Equation Exercise 16 : Power

Biophysics of Neural Computation : Introduction to Neuroinformatics
Prof. Rodney Douglas, Kevan Martin, Hans Scherberger, Matthew Cook
Ass. Frederic Zubler [email protected]
http://www.ini.uzh.ch/∼fred/bnc.html
WS 2008-2009
Exercise 15 : Cable Equation
The membrane potential V (x, t) is determined by solving the following partial differential equation (cable equation) :
τ
where:
∂v
∂t
=λ
2
∂2v
∂x2
− v + rm ie
τ = (cm rm ) sets the scale for the temporal variation in the membrane potential
λ = [(arm )/(2rL )]1/2 sets the scale for the spatial variation
a = radius of the axon (= 2 µm)
v = V − Vrest
rm = specific membrane resistance (= 1 MOhm mm2)
rL = longitudinal resistance (= 1kOhm mm)
ie = the current injected into a cell
We now assume an infinite cable and inject a constant current ie locally at x = 0. If we wait for the system to
reach its steady state ( so that ∂v
∂t = 0 ), and with some boundary conditions, the cable equation has the solution:
v(x) =
ie Rλ
2
e−|x|/λ , where Rλ =
rL λ
πa2
1. The ratio of the membrane potential at the injection site to the magnitude of the injected current is called
the input resistance of the cable. Can you express this value with the help of the last formula ?
2. We consider an infinitely long axon, which can be assumed passive, apart from two nodes where there
are sodium channels. These nodes are at x = 0 and x = 2λ. Let the resting potential be -70mV and
the threshold for action potential generation at the nodes be -50mV. Artificially we generate a prolonged
"action potential" at x = 0 (sufficiently long to assume constant current injection) of peak voltage 30mV.
Will this procedure trigger an action-potential at the second node?
3. If we double the radius of the axon, does this change the answer to question 2 ?
4. In a second axon extra membrane tightly surrounds the axon between the nodes. As a consequence the
membrane resistance increases by a factor of 4, while the capacitance decreases by a factor of 4. In all
other respects the axon is similar to the first one, with radius of 2 microns. Again, does this change
the answer to question 2 ?
Exercise 16 : Power-law for axon diameter at branch point
Studying the resistivity of dendrites, Rall came up with the now famous 3/2 power law (Rall 1959), that says
that in dendrites, the diameter of a parent segment, d0 , is related to the diameters of its daughter segments,
d1 and d2 , as (d0 )e = (d1 )e + (d2 )e , where the branch power parameter e is equal to 3/2 (cf Fig 1).
In axons, experimental data could be consistent with a similar law. But what would be the physiological justification for the existance of such a power law?
With Chklovskii and Stepanyants (BMC, 2003), we reckon that :
1. Thicker axons conduct action potential faster → it’s better to have a big diameter d.
2. Thick axons are costly, because they require large amount of cytoplasm, and occupy space → it’s better
to have a small diameter d.
These two requirements are conflicting. To tackle this problem, we try to minimize C, the combined cost of
conductance delay T and volume V :
C = aT + bV
where:
a and b are unknown constant coefficients
V = πLd2 /4 is the volume of an axon branch
T = L/(kd) is the time delay (L = length, k = constant factor for non myelinated axons)
1. Find the axon diameter that minimizes the cost C (Hint: set ∂C/∂d = 0 and solve for d).
2. The parameters a and b are still unknown, but we can use the same method to make a prediction about the
relationship among branch parameters. If the cost function for a bifurcation consisting of three segments
is : C = a1 (T0 + T1 ) + a2 (T0 + T2 ) + b(V0 + V1 + V2 ) , with
Vi = πLd2i /4 the volume of the segment number i
Ti = L/(kdi ) the time delay along the ith segment
a1 , a2 the relative costs of conduction for delays for synapses on the daughter branches.
We can group together the terms corresponding to the same segment:
C = [(a1 + a2 )T0 + bV0 ] + [a1 T1 + bV1 ] + [a2 T2 + bV2 ]
What are the segment diameters d1 , d2 and d3 which minimize the cost C ? (Hint : each term
in the latest equation depends on the diameter of only one segment. Set ∂C/∂di = 0 for i = 1, 2, 3).
3. Do these diameters satisfy the branching law (d0 )e = (d1 )e + (d2 )e ? What is the value of e ?
4. Can you think of any other branching structure in our body ?
Figure 1: Diameters at a branching point
Exercise 17 : Review on Action Potential
Assume a cell whose membrane is permeable to calcium and potassium. It shows a resting potential (RP) and
an action potential (AP) as indicated in the Fig 2.
If one varies the concentration of extracellular potassium [K+ ]out only, or the concentration of extracellular
Calcium [Ca++ ]out only, one observes the changes showed in Fig 3.
Assume that the membrane is normally permeable only to K+ ions, Ca++ ions and water. Answer the following
questions:
1. Which of these is true in the resting state (P stands for permeability): PK > PCa , PK = PCa or PK < PCa .
Explain why.
2. Same question for the peak of the AP.
Figure 2: An Action potential
Figure 3: Effect of single ion extracellular concentration changes
3. Compare [K+ ]in to [K+ ]out . Compare also [Ca++ ]in with [Ca++ ]out . In which direction do more ions
diffuse to? What is the resulting current flow? Discuss how potassium and calcium currents relate to each
other.
4. When the cell is perturbed mechanically, the membrane transiently hyperpolarizes. What permeabililty
change(s) might be responsible? Explain. What ionic currents typically lead to hyperpolarization at
physiological concentrations?
Exercise 18 : Equivalent circuit for a Synapse
Fig 4 depicts a simplified neuron, represented by an equivalent circuit including a fast chemical synapse. Vrest
is the resting potential of the cell, R is the input resistance and Vm the membrane potential. By Ohm’s law
you obtain the postsynaptic current due to a single synapse (left hand side of the diagram):
Isyn = gsyn (t) (Vm (t) − Esyn )
Note, that the synaptic conductance is a function of time, which depends on the opening of the synaptic ionic
channels. The complete equation for the above circuit (by Kirchhoff’s law) yields:
0 = Isyn + Irest = gsyn (t) (Vm (t) − Esyn ) + (Vm (t) − Vrest ) /R
Assume gsyn to be switched "on" at t = 0, so that
potential arrives at the synapse) and is switched "off"
from the synaptic cleft) such that:


0
gsyn (t) = 1nS


0
Assume Vrest = -70mV and R = 2GOhm
current may pass through (when a presynaptic action
at t = 1ms (when all neurotransmitter has been cleared
for t < 0,
for 0 < t 6 1ms,
for t > 1ms
1. Let Esyn = 10mV. Is this a excitatory or inhibitory synapse? Calculate Isyn and the amplitude of the
postsynaptic potential (i.e. Vm for 0ms < t < 1ms).
Biophysics of Neural Computation, Introduction to Neuroinformatics (Winter 2006
2. Let Esyn = -90mV. Is this a excitatory or inhibitory synapse? Calculate Isyn and the amplitude of the
postsynaptic potential.
3. Let Esyn = -70 mV. Calculate Isyn and the amplitude of the postsynaptic potential. Calculate gtotal (total
Question
membrane conductance). What
effect1 will this synapse have on the responsiveness of the membrane? Is
this a excitatory or inhibitory synapse?
The figure below depicts a simplified neuron, represented by an equivalent circuit i
4. What circuit element can you
add to make the circuit more realistic (hint: think of the time-course of the
synapse.
responses)?
Figure 4: An equivalent circuit including a fast chemical synapse
Vrest is the resting potential of the cell, R is the input resistance and Vm the membrane po
By Ohm’s law you obtain the postsynaptic current due to a single synapse (left hand sid
Isyn = gsyn (t) * (Vm (t) - Esyn )
Exercise 19 : Pre- and post- synaptic recordings.
Question 2
in Fig 5 you see four traces of recordings from a presynaptic cell and its postsynaptic cell. Voltage and current
In the
figure
you see four
traces of recordings
from aofpresynaptic
cell and
its postsynaptic
cel
Note,
thatbelow
the synaptic
conductance
is a function
time, which
depends
on the open
are plotted on the same time scale. Identify which trace is
andchannels.
current areThe
plotted
on the same
time scale.
Identify
trace
complete
equation
for the
abovewhich
circuit
(byis Kirchhoff’s law) yields:
1. the postsynaptic current
1) the postsynaptic current
= Isyn + Irest = gsyn (t) * (Vm (t) - Esyn ) + (Vm (t) - Vrest ) / R
2. the presynaptic action potential2) 0
the presynaptic action potential
3. the postsynaptic potential
3) the postsynaptic potential
4) gthe
Ca2+-current
Assume
to be switched
‘on’ at t=0, so that current may pass through (when a pre
4. the presynaptic Ca2+-current
synpresynaptic
how
to yourand
conclusion.
arrives
atyou
thecame
synapse)
is switched ‘off’ at t = 1ms (when all neurotransmitter h
Explain how you came to Explain,
your
conclusion.
synaptic cleft) such that:
0 for t < 0
gsyn (t)
=
1nS for 0 < t <= 1ms
0 for t > 1ms
Assume Vrest = -70mV and R = 2GOhm
(a) Let Esyn = 10mV. Is this a excitatory or inhibitory synapse? Calcula
postsynaptic potential (i.e. Vm for 0ms < t < 1ms) and Isyn .
(b) Let5: ESome
-90mV. ...
Is this a excitatory or inhibitory synapse? Calcula
syn =recordings
Figure
Question 3 postsynaptic potential and Isyn .
You are designing
an experiment
test Calculate
the probabilistic
nature
neurotransmitter
You know,
(c) Let
Esyn = -70 tomV.
Isyn and
theofamplitude
of therelease.
postsynaptic
po
the probability
of (total
releasemembrane
is small (e.g.conductance).
if you lower extracellular
Ca++will
in your
and the
gtotal
What effect
this preparation)
synapse have
on nt
synaptic vesicles available for release is large (as at the neuromuscular junction) then the expected n
membrane? Is this a excitatory or inhibitory synapse?
vesicles released following a presynaptic action potential follows a Possion distribution. Therefore if
Exercise 20 : Quantal release I
You are designing an experiment to test the probabilistic nature of neurotransmitter release. You know that
when the probability of release is small (e.g. if you lower extracellular Ca++ in your preparation) and the
number of synaptic vesicles available for release is large (as at the neuromuscular junction) then the expected
number of vesicles released following a presynaptic action potential follows a Possion distribution. Therefore if
m is the mean number of vesicles released per trial, then the probability p of observing a particular number x
x
(x = 0, 1, 2, 3 ...) of vesicles released in a given trial is: px = mx! e−m
You stimulate a nerve 500 times. Assume that each trial is independent of the other trials performed. Given
that the number of quanta released in one trial follows Possion statistics and the mean quantal content is 5.
1. How many failures of transmission do you expect to observe in the 500 trials?
2. How often do you expect two quanta to be released ?
Exercise 21 : Quantal release II
Fig 6 shows a histogram of end-plate potential of a cat muscular fiber. These data were recorded in 1956 by
I. A. Boyd and A. R. Martin to study the release properties of the muscular end-plate. The recording were
achieved under low extracellular Ca++ and high Mg++ concentrations. These special conditions permitted
to reach extremely low release probabilities. This make up the end-plate potential to fluctuate according to
Poisson’s law (see previous exercise).
The histogram of Fig 6 corresponds to the amplitude distribution of 200 events (evoked and spontaneous). The
mean amplitude of all events is 0.93 mV.
1. What do the different peaks labelled A, B, C, and D correspond to?
2. From the graph what is Q, the quantal amplitude if exactly one vesicle is released.
3. Using the result from b) and the mean amplitude of all events calculate m, the mean number of vesicles
released per trial. Can you imagine another way of calculating m assuming a Poisson distribution?
4. Assume that m and Q are measured for a given experiment. In a subsequent measure in which experimental
parameters have changed, m and Q are measure again and they might have changed. How can you detect
an increase in the release probability? How can you detect postsynaptic modifications?
5. It is possible to fit a Gaussian to each peak of the distribution and then add them to obtain the theoretical
distribution. In order to do that you need to know how many events to include under each curve. What
do these number of events correspond to and how would you calculate them.
Exercise 22 : Inhibitory synapses
The most important sources for inhibition in the central nervous system are mediated by GABA and glycine
receptors. These channels are selective for anions, such as Cl− or HCO−
3 . For most neuronal cell bodies,
the equilibrium for anions is more negative then the resting potential (because Cl− is pumped out of the cell).
However, in some cells, the equilibrium for ions is sometimes more positive than the resting membrane potential,
and the opening of GABA or glycine channels produces a depolarization.
1. In this case, how could the effect still be inhibitory?
2. The figure 7 shows the effect of an unknown peptide that opens such a receptor-channel at different
voltages (the y-axis represents the difference of current if you release the peptide). Remember that by
definition a flux of positive charge going outward is a positive current. If the ions flowing through
this channel (when it is open) are negative (anions), do they go outward or inward at the
resting potential (-70 mV)?
3. what is the reverse potential for this channel?
4. If this peptide was designed to be an anti-epileptic drug, would it be a good candidate?
Figure 6: End-plate potential of a cat muscular fiber
Figure 7: Effect of our mysterious peptide at different voltages (note that the y axis shows the difference of current with
the peptide)
Exercise 23 : Spike time dependent plasticity
It is experimentally observed that the relative timing of pre-and post-synaptic action potential can change the
synaptic efficacy. Within a time window of 50ms, presynaptic spike that precede postsynaptic action potential
produce LTP (strengthening of the synapse), whereas presynaptic spikes that follow postsynaptic AP produce
LTD (weakening of the synapse).
1. Why is it important -for any learning rule- to have mechanisms for both the strengthening
and the weakening of the synaptic weight?
2. The figure 8 shows the spike-time-dependent plasticity (STDP) modification function for a particular
synapse.
F (∆t) is the amount of synaptic weight modification observed after a pair of pre- and postsynaptic spikes,
as function of ∆t = tpre − tpost .
(
F (∆t) =
A+ exp(∆t/τ + ) if ∆t < 0
A− exp(∆t/τ − ) if ∆t > 0
Estimate the value of the parameters A+ and tau for the function F of Fig 8.
Figure 8: Spike-time-dependent plasticity (STDP) modification function for a particular synapse, Song & al. (2000)
3. If two neurons fire completely independently, there is on average as much occurrences where the first one
spikes just before the second one does, than cases where the opposite happens. Nevertheless, a synapse
between two such neurons should be weakened, which will only happen if the integral of the function F is
negative, i.e. if A+ τ + < A− τ − . Draw a weight update function (similar to Fig 8) that ensures
that uncorrelated pre- and postsynaptic spike produce an overall weakening of the synapse.
4. Such an STDP rule would tend to balance the firing rate in a neuron. As state Song & al. (2000) :
"(...) a neuron can operate in two different modes with distinct spike-train statistics and input/output
correlations. When excitation is strong (...) so that the mean input to the neuron would bring it well
above threshold if action potentials were blocked, the neuron operates in an input-averaging or regularfiring mode. The postsynaptic action potential sequences produced in this mode are significantly more
regular than the presynaptic spike trains that evoke them. The interspike intervals of the postsynaptic
response depend on the total synaptic input, but the absolute timing of individual postsynaptic action
potentials is fairly insensitive to presynaptic spike times. As a result, there are roughly equal numbers of
presynaptic action potentials before and after each postsynaptic spike". And in this case, if the integral
of F is negative, we have a weakening of the synapse.
"As the excitatory synapses are weakened by STDP, the postsynaptic neuron enters a balanced mode
of operation in which it generates a more irregular sequence of action potentials that are more tightly
correlated with the presynaptic spikes that evoke them. The total synaptic input in the balanced mode
is, on average, near or below threshold, so the postsynaptic neuron fires irregularly, primarily in response
to statistical fluctuations in the total input. Because action potentials occur preferentially after a random
fluctuation, there tend to be more excitatory presynaptic spikes before than after a postsynaptic response".
What happends then?
Exercise 24 : Dendritic computation
An excitatory synapse increases locally the membrane potential when activated. This depolarization
spreads passively toward the soma. An inhibitory "shunting", or "silent" synapse can fix the potential
locally to the resting potential. It can be seen as a veto on any more distal synapse, whose membrane
potential change will be stopped. But an inhibitory synapse has little effect in vetoing a more proximal
exitatory synapse.
(a) In Fig 9, which combination has a chance to elicit an action potential in the axon?
i. e1 + i7 activated, all the others inactivated
ii. e2 + i1 + i4 + e5 activated, all the others inactivated
iii. all the synapses are activated
(b) What synaptic conditions must exist for e6 to have an influence on the membrane
potential at the beginning of the axon?
Figure 9: Schematic dendritic tree of a cell. The excitatory synapses are e1, e2, ..., e6 , and the inhibitory ones i1, ...,
i7. Figure taken from [Koch and Shepherd, 1990, in The synaptic organization of the brain , Oxford University Press].
(c) If all the inhibitory synapses have the same probability to be active, which excitatory
synapse will have more influence on the membrane potential at the beginning of the
axon, e2 or e5?
(d) What is the number of possible combinations of activation for the synapses on this
neuron?
(e) If this neuron needs at least the effect of 2 excitatory synapses on the beginning of the axon to reach
the firing threshold, give a possible combination that meets this condition (write the required
activation condition for each synapse, ie active or not).
Exercise 25 : Interaural time differences
(a) Look at Fig 10. Describe the nature of the spike input from the left and right side in
order to process inter-aural time differences. Is it a temporal or a rate code? Explain.
(b) The neurons (A-E) serve as coincidence detectors and fire maximally when they receive the left
and right sided input simultaneously. Nevertheless, the output of the neurons (A-E) has different
tuning curves, due to the different lengths of the axons from the left and right sided input neurons.
Describe the tuning properties of the output neurons (A-E) qualitatively.
(c) Given that the speed of sound in the air is 350 m/s and the distance between the left and right
ear is 15 cm (in humans), how big is the time delay between the left and right ear for a
sound coming directly from the side,? How does that relate to the duration of an action
potential?
Exercise 26 : Reading a neuron’s specificity
Fig 11 shows spike rasters and peristimulus time histograms of a parietal cortex neuron for reach (blue)
and eye movements (red) in 8 center-out directions. (For the saccade task, eye position traces are shown
at the bottom of each panel).
(a) Describe the spiking activity of this neuron for the two behaviors, movement directions,
and in within the task (baseline, cue, planning, execution phases). What is this neuron
coding for?
(b) What can be said about the activity variation from trial to trial?
Figure 10: Coincidence detectors for sound localization.
Figure 11: Raster plots and PSTH for reach (blue) and eye movements (red) in 8 center-out directions.