1. Synapses and Passive Cable Theory Methods in Computational Neuroscience

2. A mathematical model of a neuron
Equivalent circuit model
gNa gK gL EL EK
ENa Ie V C +
Alan Hodgkin
Andrew Huxley, 1952

3. Synaptic inputs onto dendrites
So far we have considered a very simple model of neurons – a model consisting of a single capacitor representing the soma of the neuron.
We also treated ‘input’ to a neuron as current injection into the soma.
But in a real neuron, inputs come onto neurons through synapses, and most synaptic inputs arrive onto dendrites far from the soma.
Ramon y Cajal

4. Synaptic inputs onto dendrites
How does a synapse affect the membrane potential of a postsynaptic neuron?
How does input onto a dendrite affect membrane potential at the some? - +

5. Sequence of events at a chemical synapse
+50mV
-60 mV
voltage-gated Ca- channels
Ca++
dendrite

6. Sequence of events at a chemical synapse
‘ligand’ = ‘neurotransmitter’
Ca++
Ca++
ligand-gated ion channels
Na+
Ca++
dendrite

7. How does the postsynaptic neuron respond?
Ionotropic receptors
Two electrode voltage-clamp experiment
Magleby and Stevens, 1972 - +
Frog sartorius muscle fiber
Motor neuron synapse

8. How does the postsynaptic neuron respond?
I-V Curve
50 mV
-100
1 ms
time

9. Equivalent circuit model of a synapse
Current flow through a synapse results from changes in synaptic conductance
Esyn
Gsyn
Equivalent circuit of a synapse Ei Ie V C + Gi

10. Excitatory and inhibitory synapses
Increased synaptic conductance causes the membrane potential to approach the reversal potential for that synapse.
Gsyn Ei Ie V C + Gi
Excitatory synapse if
Excitatory
postsynaptic
potential (EPSP)

11. Excitatory and inhibitory synapses
Increased synaptic conductance causes the membrane potential to approach the reversal potential for that synapse.
GABAergic synapse
Gsyn Ei Ie V C + Gi
Inhibitory synapse if
Inhibitory
postsynaptic
potential (EPSP)

12. Signal propagation in dendrites and axons
How does a pulse of synaptic current affect the membrane potential at the soma? - +

13. Injected current per unit length
We are going to take a simplified view of dendrites - a thin insulating cylinder with conductive solution.
Radius a
axial
Injected current per unit length
times segment length
inside
outside

14. Ohm’s Law in a cable Let’s write down the relation between
V(x,t) and I(x,t)
Radius a
axial
Ohm’s Law
inside
outside
Note that a negative gradient of V produces a current flow to the right
This is just the definition of a derivative!

15. Ohm’s Law in a cable Consider the special case of a length L
Radius a
And no membrane conductances…
inside
open end
If there are no membrane conductances then:
Membrane potential changes linearly! V0 L
Integrate over x:

16. Input Impedance
Input impedance (resistance) is simply the ratio of voltage change to injected current.
inside
open end V0 L

17. Ohm’s Law in a cable Consider the special case of a length L
Radius a
And no membrane conductances…
inside
Now consider the ‘sealed end’ boundary condition
What is the input impedance? V0 L

18. Deriving the cable equation
Injected current per unit length
inside
Kirchoff’s law: sum of all currents out of each node must equal zero.
Membrane current per unit length
Length of element
But remember that:
Assuming ra is constant
Substitute

19. Deriving the cable equation
This we know!!
Each element in our cable is just like our model neuron!
So, the total membrane current in our element of length Δx is:
Capacitance per unit length
Membrane ionic conductance per unit length
Plug this expression for into the equation at top…

20. Deriving the cable equation
Divide both sides by Gm to get the cable equation!
Steady state space constant (length, mm)
where
Membrane time constant (sec)
Specific membrane resistance (Ω mm)

21. EL is just a constant offset
An example
Let’s solve the cable equation for a simple case. What is the steady state response to a constant current at a point in the middle of an infinitely long cable?
EL is just a constant offset
so we ignore it

22. An example
Fix current curve. Negative for x<0

23. An analogy
Lets look at this in more detail: Why does an exponential fall off in current make sense?
Leaky garden-hose analogy
Current is like water flow
Voltage is like pressure
Thus, the dendrite acts like a series of current dividers.

24. Input impedance of semi-infinite cable
We can calculate the input impedance
We calculated earlier that the current along the cable is
If we evaluate the current at x=0, we get:
Thus the ‘input impedance’ of a cable is just the axial resistance of a length λ of the cable!
Thus,
What can we say about the input conductance?
since

25. Another look at units Total membrane conductance : total area
conductance per unit area (S/mm2)
Membrane conductance per unit length :
circumference
Units are S/mm
Specific membrane resistance:

26. Another look at Units
Axial resistance: the resistance along the inside of the dendrite
Total axial resistance along a dendrite of length
Resistivity of the intracellular space (property of the medium )
where
A = cross sectional area =
Axial resistance per unit length
Electrotonic length
Depends on radius, membrane conductance and resistivity of intracellular space

27. Typical λ for a dendrite of a cortical pyramidal cell
First calculate membrane resistivity
Resistivity intracellular space
Now we calculate axial resistance

28. Scaling with radius
Neurons need to send signals over a distance of a ~100 mm in the human brain.
What would a (radius) would have to be to get λ= 100 mm?
This would never work! This is why signals that must be sent over long distances in the brain are sent by propagating axon potentials.

29. Electrotonic length
Electrotonic length is the physical length divided by the space constant.
unitless
The amount of current into the soma will scale as

30. Pulse of charge
We can exactly solve the case of a brief pulse of current in an infinite cable
where

31. Pulse of charge Pulse of charge video
Looking at just the spatial dependence
Pulse of charge video
This is just a Gaussian profile.

32. Pulse of charge Width increases as
Note that in a time 0.1τ, the charge relaxes to a spatial extent of λ
Looking at just the spatial dependence
Width increases as

33. Pulse of charge Using , we can rewrite this as
Units are charge per unit length
In the formulation, Q(X,T) is charge density, which you get by multiplying by Cm (capacitance per unit length).
But in electrotonic units, C(lambda) is already capacitance per unit length (capacitance per lambda).
Thus, the total charge leaks away as

34. Pulse of charge
The scale of the voltage change (from charge Q) is set of the capacitance of a segment of length λ:
We saw earlier that by very short times (T=0.1) the charge relaxes from a δ-function to a spread of λ
But the capacitance of the membrane holding this charge is just
Therefore, the voltage is given by

35. Propagation Find the peaks by calculating and setting it to zero.- +
Find the peaks by calculating
and setting it to zero.
For any given X, you can solve for Tmax.
From this, we can calculate the velocity!

36. Dendritic filtering
As the voltage response propagates down a dendrite, it not only falls in amplitude, but it broadens in time. - +

37. Compartmental models soma dendrite Somatic compartment
Dendritic compartment
Somatic compartment
Dendritic compartment