Single neuron modelling

1
Single neuron modelling

• Computational Neuroscience 03
• Lecture 2

2
Contents

• Electrical properties of neurons (membrane
  equation, Nernst equation etc)
• Simple models
• Adding conductances
• Hodgkin-Huxley model
• Other considerations

3

• 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.

4
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. Dont 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)
5
Most channels are highly selective for a
particular type of ion Capacity 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/neuromodulators Also 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)
6
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
7
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 tm Rm Cm rm cm
sets the basic time-scale for changes in the
membrane potential (typically between 10 and
100ms)
8
Nernst equn and equilibrium potential
Potential 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
9
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 gt EK K ions will flow out of
neuron and so hyperpolarise it Conversely, as Na
and Ca have ve Es normally V lt E and so ions
flow in and depolarise neuron
10
Membrane current
The 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 S
gi(V - Ei) Conductances change over time leading
to complex neuronal dynamics.
11
However have some constant factors (eg current
from pumps) which are grouped together as a
leakage current.
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
12
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
By convention electrode current is ve inward
while membrane current is ve outward
13
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 ltVth
• 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

14
passive integrate fire
Simplest model is a passive model which assumes
NO active conductances. Therefore
so
Multiplying through by rm1/gL we get
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
15
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
16
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)
17
Adding a conductance
We 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 -gt gsra Dgsra
18
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
19
Voltage-dependent Conductances
Most of the interesting neuronal dynamics arise
from the dynamics of active conductances Recording
s show that channels have stochastic opening and
closing depending on V, transmitter and Ca (Well
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
20
2 types of channel persistent (A) and transient
(B)
21
Persistent gates

• Pi increases for depolarisation decreases for
  hyperpolarisation
• Pnk where k is no. of independent events that
  must happen (eg potassium gated channel needs 4
  things to happen so k4). In practice, integer
  changed to fit data (HH didnt know physiology
  but suggested 4).
• n is probability that an individual sub unit is
  open
• Closed -gt open at rate a(V), open-gt closed at
  rate b(V)
• Thus
• dn/dt an(V) (1-n) - bn(V) (n)
• Alternatively

22
For a voltage activated conductance,
depolarisation makes n grow and hyperpolarisation
makes it shrink. Therfore expect an to be
increasing in V and bn to be decreasing in V Use
thermodynamic arguments to get a general form for
an and bn 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)
23
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 am , bm and ah
  , bh
• 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
24
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
These equations can be integrated numerically Can
also be formulated in multi-compartment model
which shows how APs propagate along axons
25
Initial rise in V is due to current injected at
t5ms 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
26
Extensions
Channels 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)
27
Phase plane analysis
As 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
Dt), w(tDt)) which, if Dt is sufficiently small
is in the direction of du/dt or dw/dt
Eg where uvE and vvI
28
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