Single neuron modelling

Computational Neuroscience 03

Lecture 2

Contents

• Electrical properties of neurons (membrane equation, Nernst equation etc)

• Simple models

• Adding conductances

• Hodgkin-Huxley model

• Other considerations

Neurons are enclosed by a membrane separating interior from extracellular space

The concentration of ions inside is different (more –ve) to that in the surrounding liquid

-ve ions therefore build up on the inside surface of the membrane and an equal amount of +ve ions build up on the outside

The difference in concentration generates an electrical potential (membrane potential) which plays an important role in neuronal dynamics.

Cell membrane: 2-3 nm thick and is impermeable to most charged molecules and so acts as a capacitor by separating the charges lying on either side of the membrane(NB Capacitors store charge across an insulating medium. Don’t allow current to flow across, but charge can be redistributed on each side leading to current flow)The ion channels in the membrane lower the effective membrane resistance by a factor of 10,000 (depending on density, type etc)

Most channels are highly selective for a particular type of ionCapacity of channels to conduct ions can be modified by eg membrane potential (voltage dependent), internal concentration of intracellular messengers (Ca-depdt) or external conc. Of neurotransmitters/neuromodulatorsAlso have ion pumps which expend energy to maintain the differences in concentrations inside and outside

Exterior potential defined to be 0 (by convention). Because of excess –ve ions inside, resting membrane potential V (when neuron is inactive) is –ve.

Resting potential is the equilibrium point when ion flow into the cell is matched by ion flow out of cell

V will vary at different places within the neuron (eg soma and dendrite) due to the different morphological properties (mainly the radius)

Membrane capacitance and resistance

Neurons without many long narrow cable segments have relatively uniform membrane potentials: they are electrotonically compact

Start by modelling these neurons with assumption that membrane potential is constant: single compartment model

Denoting membrane capacitance by Cm and the excess charge on the membrane as Q we have: Q = CmV and dQ/dt = CmdV/dt

Shows how much current needed to change membrane potential at a given rate

Membrane also has a resistance: Rm Determines size of potential difference caused by input of current: IeRm

Both Rm and Cm are dependent on surface area of membrane A.

Therefore define size-independent versions, specific membrane conductance cm and specific membrane resistance rm where Cm = Acm etc

Membrane time constant m = Rm Cm = rm cm sets the basic time-scale for changes in the membrane potential (typically between 10 and 100ms)

Nernst equn and equilibrium potentialPotential difference between outside and inside attracts +ve ions in and repels –ve ions out

Difference in concentration between inside and outside mean ions diffuse through channels (Na+ and Ca2+ come in while K+ goes out)

Define equilibrium potential E for a channel as membrane potential at which current flow due to electric forces cancels diffusive flow

Eg Consider +ve ion and –ve membrane potential V: V opposes ion flow out, so only those with enough thermal energy can cross the barrier so at equilibrium get:

[outside] = [inside] exp(zE/VT) where z is no. of extra protons of ion, VT is a constant (from thermal energy of ions) and E is equilibrium potential

Solve to get Nernst Equation:

From Nernst equation get equilibrium potentials of channels:EK is typically between –70 and –90 mV, ENa is 50mV or higher, Eca is around 150mV while Ecl is about 65mV (near resting potential of many neurons)

A conductance with an equilinrium potential E tends to move membrane potential V towards E eg if V > EK K ions will flow out of neuron and so hyperpolarise it

Conversely, as Na and Ca have +ve E’s normally V < E and so ions flow in and depolarise neuron

][

][ln

inside

outside

z

VE T

Membrane currentThe membrane current is total current flowing through all the ion channels

We represent it by im which is current/unit area of membrane

Amount of current flowing each channel is equl to driving force (difference between equilibrium potential Ei and membrane potential) multiplied by channel conductance gi

Therefore:

im = gi(V - Ei)

Conductances change over time leading to complex neuronal dynamics.

However have some constant factors (eg current from pumps) which are grouped together as a leakage current.

)( LL EVg

Overline on g shows that it is constant. Thus it is often called a passive conductance while others termed active conductances

Equilibrium current is not based on any specific ion but used as a free parameter to make resting potential of the model neuron match the one being studied

Similarly, conductance is adjusted to match the membrane conductance at rest

Single compartment model

This is the basic model for all single compartment models.

Rate of change of the membrane potential is proportional to rate at which charge builds up inside cell = current entering into neuron

Current in = membrane current + external current from electrode

Therfore, using size-independent variables have:

A

Ii

dt

dVc e

mm By convention electrode current is +ve inward while membrane current is +ve outward

Integrate and fire models• These models basically assume that action potentials are

simply spikes ocurring when the mebrane potential reaches a threshold Vth

• After firing membrane potential is reset to a Vreset <Vth

• Simplifies the modelling dramatically as we only deal with subthreshold membrane potential dynamics

• Can be modelled at various levels of rigour depending on simplifying assumptions used

passive integrate +fireSimplest model is a passive model which assumes NO active conductances. Therefore:

)( LLm EVgi A

IEVg

dt

dVc e

LLm )(so

Multiplying through by rm=1/gL we get:

emLm IRVEdt

dV

And if V reaches Vth an AP is fired after which V is reset to Vreset

If Ie is 0 V decays exponentially with time constant tm to EL

These equations can be solved numerically for different forms of current as in the figure above

Numerical integration techniques can be found in eg Methods in Neuronal modelling, Numerical Recipes in C and appendices of Theoretical neuroscience

However, for constant current it can be solved directly

This leads to the prediction that the firing rate is a linear function of current (fig A above).

However, while the model fits data from the inter-spike intervals from the first 2 spikes well, it cannot match the spike rate adaptation which occurs in real neurons

For this to occur, we need to add an active conductance (fig C)

Adding a conductanceWe now want to add an active conductance gsra to the model to allow for spike rate adaptation but retain as much simplicity as possible

gsra based on K+ so that when activated ions will flow out and hyperpolarise neuron and slow spiking

Dynamics of gsra : with no spikes decays to 0 exponentially (so first 2 spikes are OK. If spike: gsra -> gsra + gsra

emKsramLm IREVgrVEdt

dV )(

Does not capture refractory dynamics however (lower probability of neuron firing shortly after a spike)

Could add this via simply banning firing for short period after firing, or adding a conductance with a faster time decay and larger conductance increment, or raise AP threshold post-spike etc

Ie make model more complicated and dynamic

Voltage-dependent ConductancesMost of the interesting neuronal dynamics arise from the dynamics of active conductances

Recordings show that channels have stochastic opening and closing depending on V, transmitter and Ca (We’ll start with V and do Ca later)

Most models are deterministic due to large number of channels with conductance given as a fraction of maximal conductance (depends on channel type and density)

Probability of finding a channel open is Pi(V or Nt or Ca)

Here we will discuss (following HH) the delayed rectifier K+ and fast Na+ channels

2 types of channel: persistent (A) and transient (B)

Persistent gates• Pi increases for depolarisation decreases for hyperpolarisation• P=nk where k is no. of independent events that must happen (eg

potassium gated channel needs 4 things to happen so k=4). In practice, integer changed to fit data (HH didn’t know physiology but suggested 4).

• n is probability that an individual sub unit is open• Closed -> open at rate (V), open-> closed at rate (V)• Thus• dn/dt = n(V) (1-n) - n(V) (n)Alternatively:

)()(

)()(:

)()(

1)(:

)()(

VV

VVnand

VVVwhere

nVndt

dVV

nn

n

nnn

n

For a voltage activated conductance, depolarisation makes n grow and hyperpolarisation makes it shrink. Therfore expect n to be increasing in V and n to be decreasing in V

Use thermodynamic arguments to get a general form for n and n

and then use experimental data to fit them as shown by solid lines above (dotted lines are general forms adjusted to fit to data as used by HH)

Transients Channels• Same as before but now to be open have to be open and

not inactivated PNa =m3h

• Now have 2 sets of rate constants m , m and h , h

• Fit to data using same generic equations as before: now need to hyperpolarise to raise h then depolarise

Channels also exist ones with just a ball: open when hyperpolarised

Hodgkin-Huxley model

HH model in single compartment form now adds a persistent K and transient Na channel to the simple leakage model we had earlier:

)()()( 34NaNaKKLLm EVhmgEVngEVgi

These equations can be integrated numerically

Can also be formulated in multi-compartment model which shows how APs propagate along axons

Initial rise in V is due to current injected at t=5ms which drives current up to about –50mV

At this point m rises sharply to almost 1 while h is also, tansiently, non-zero.

This causes an influx of Na+ ions and a large rapid depolarisation to about 50mV due to +ve feedback because m increases with V

However, increasing V causes h to decrease shutting off Na current

Also, n increases activating K+ channels and ion flow outward

Finally, values return to initial values

ExtensionsChannels can also be modelled stochastically wthough HH is a good approx for 100+ channels

Can also add many other ion channel dynamics

Can also add synaptic channels to both this model and integrate and fire model

Can add more compartments (ie reintroduce morphology)

Phase plane analysisAs you have no doubt noticed it is not easy to visualise the behaviour of the parameters in the HH model

A common technique for doing this is to use a phase-plane analysis where we look at the temporal evolution of 2 parameters (u, w).

Idea is that from a starting point (u(t), w(t)) the systme will move in time to a new state (u(t+ t), w(t+t)) which, if t is sufficiently small is in the direction of du/dt or dw/dt

Eg where u=vE and v=vI

Places where dv/dt or du/dt = 0 are called null-clines

Can then plot arrows over all the plane to see what the behaviour of the system is likely to be

Characterise the long term behaviour by looking for attractors (fixed points or limit cycles)

Obviously, need to reduce model to 2 dimensions

2 more egs of phase plane diagrams