Synapses and Passive Cable Theory Methods in Computational Neuroscience

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

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

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? - +

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

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

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

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

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

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)

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)

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

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

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!

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:

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

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

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

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…

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)

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

An example Fix current curve. Negative for x<0

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.

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

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:

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

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

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.

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

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

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

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

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

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

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!

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

Compartmental models soma dendrite Somatic compartment Dendritic compartment Somatic compartment Dendritic compartment