Passive Cable Theory Methods in Computational Neuroscience 1

A mathematical model of a neuron Equivalent circuit model Alan Hodgkin Andrew Huxley, g Na gKgK gLgL ELEL EKEK E Na IeIe V C + + +

Two electrode voltage-clamp experiment Magleby and Stevens, Frog sartorius muscle fiber Motor neuron synapse 3 How does a synapse respond? Ionotropic receptors

How does a synapse respond? Ionotropic receptors 50 mV ms time I-V Curve 4

Equivalent circuit model of a synapse Current flow through a synapse results from changes in synaptic conductance EiEi IeIe V C + GiGi E syn G syn Equivalent circuit of a synapse

Excitatory and inhibitory synapses Increased synaptic conductance causes the membrane potential to approach the reversal potential for that synapse. G syn EiEi IeIe V C + GiGi Excitatory synapse if Excitatory postsynaptic potential (EPSP) 6

Increased synaptic conductance causes the membrane potential to approach the reversal potential for that synapse. G syn EiEi IeIe V C + GiGi Inhibitory synapse if Inhibitory postsynaptic potential (EPSP) GABAergic synapse 7 Excitatory and inhibitory synapses

Signal propagation in dendrites and axons 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 did this because in most vertebrate neurons, the region that initiates action potentials is at the soma. This is usually where the ‘decision’ is made in a neuron whether to spike or not. 8 Ramon y Cajal

However relatively few inputs to a neuron are made onto the soma. Most synaptic inputs arrive onto the dendrites – which are thin branching processes that radiate from the soma. Synapses onto the dendrite effectively inject current into the dendrite at some distance from the soma (as much as 1-2 mm away) How does a pulse of synaptic current affect the membrane potential at the soma? - + Signal propagation in dendrites and axons 9

Radius a Injected current per unit length times segment length outside We are going to take a simplified view of dendrites - a thin insulating cylinder with conductive solution. inside axial 10

Model assumptions Ignore magnetic fields- no EM propagation. We only model current flow parallel to the axis. Extracellular resistance is small compared to intracellular. –Geometric argument: dendrites are small. Extracellular currents extend over a wide area reducing the resistance. Radius a 11

outside inside Ohm’s Law in a cable Ohm’s Law Let’s write down the relation between V(x,t) and I(x,t) Note that current flow to the right produces a negative gradient Radius a axial This is just the definition of a derivative! 12

If there are no membrane conductances then: Membrane potential changes linearly! inside Radius a And no membrane conductances… Ohm’s Law in a cable Consider the special case of a length L Integrate over x: V0V L

Input Impedance 14 V0V0 0 Input impedance (resistance) is simply the ratio of voltage change to injected current. L inside What would the input impedance be with open end?

inside Deriving the cable equation Injected current per unit length 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: Substitute Assuming r a is constant 15

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 This we know!! Deriving the cable equation Plug this expression for into the equation at top… 16

Membrane time constant (sec) Specific membrane resistance (Ω mm) Divide both sides by G m to get the cable equation! Steady state space constant (length, mm) where Deriving the cable equation 17

Another look at units Total membrane conductance : total area conductance per unit area (S/mm 2 ) Specific membrane resistance: Membrane conductance per unit length : circumference Units are S/mm 18

Axial resistance: the resistance along the inside of the dendrite Axial resistance per unit length Electrotonic length Another look at Units Resistivity of the intracellular space (property of the medium ) where Total axial resistance along a dendrite of length A = cross sectional area = Depends on radius, membrane conductance and resistivity of intracellular space 19

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? E L is just a constant offset so we ignore it An example 20

An example 21

Lets look at this in more detail: Why does an exponential fall off in current make sense? An analogy Leaky garden-hose analogy Thus, the dendrite acts like a series of current dividers. Current is like water flow Voltage is like pressure 22

We can calculate the input impedance since Input impedance of semi-infinite cable We calculated earlier that the current along the cable is If we evaluate the current at x=0, we get: Thus, Thus the ‘input impedance’ of a cable is just the axial resistance of a length λ of the cable! What can we say about the input conductance? 23

Typical λ for a dendrite of a cortical pyramidal cell First calculate membrane resistivity Resistivity intracellular space Now we calculate axial resistance 24

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. Scaling with radius 25

Electrotonic length Electrotonic length is the physical length divided by the space constant. unitless The amount of current into the soma will scale as 26

We can exactly solve the case of a brief pulse of current in an infinite cable Pulse of charge where 27

Pulse of charge Looking at just the spatial dependence This is just a Gaussian profile. 28 Pulse of charge video

Pulse of charge Looking at just the spatial dependence Width increases as Note that in a time 0.1τ, the charge relaxes to a spatial extent of λ 29

Pulse of charge Thus, the total charge leaks away as Using, we can rewrite this as Units are charge per unit length 30

The scale of the voltage change (from charge Q) is set of the capacitance of a segment of length λ: Pulse of charge 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 31

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

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

Two-compartment models somadendrite Somatic compartment Dendritic compartment Somatic compartment Dendritic compartment 34