Computationally Modeling Neurons Lecture 3

1
Computationally Modeling Neurons(Lecture 3)

• Harry R. Erwin, PhD
• COMM2E
• University of Sunderland

2
Road map

• Introduction to Neural Modelling
• Chemical Dynamics
• Electrodynamics
• Putting it all together
• Conclusions

3
Introduction to neural modelling

• Resources consulted include
• Shepherd, The Synaptic Organization of the Brain,
  5th edition, Oxford.
• Dayan and Abbott, Theoretical Neuroscience, MIT
  Press.
• Koch, Biophysics of Computation, Oxford.
• Bower and Beeman, The Book of Genesis, 2nd
  edition, available electronically.
• Nicholls, et al., 4th edition.

4
The Platonic neuron

• Consists of a soma, one or more dendrites, and an
  axon
• Any of these elements may be missing
• You can also have binary neurons where a dendrite
  is axon-like. These can signal bi- or
  unidirectionally

Apical Dendrite
Soma
Basal Dendrite
Axon and Axon Collateral
5
A real pyramidal neuron
Beeman, 2005, Introduction to Realistic Neural
Modeling, http//www.wam-bamm.org/Tutorials/genes
is-intro/genesis-intro.html
6
A Purkinge neuron model
4550 compartments Beeman, 2005, Introduction to
Realistic Neural Modeling, http//www.wam-bamm.or
g/Tutorials/genesis-intro/genesis-intro.html
7
How we model them
Beeman, 2005, Introduction to Realistic Neural
Modeling, http//www.wam-bamm.org/Tutorials/genes
is-intro/genesis-intro.html
8
Neuron Properties

• The number of synapses can range from one
  (auditory neurons) to thousands (most
  interneurons), tens of thousands (cortical
  pyramidal cells) and hundreds of thousands
  (Purkinje cells of the cerebellum)
• The soma can range between 10 and 50 ?m in
  diameter (10-2 to 5x10-2 millimeters).

9
Spatial Scales

• In the cortex, dendritic trees run about 10 mm in
  total length. Axons run about 40 mm in total
  length.
• Motor neurons can extend from the head to the
  hand or foot. In whales, they can extend from the
  head to the tip of the fluke. In dinosaurs, they
  were as long as thirty meters, or three seconds
  from head to the end of the tail.
• This is why the spine has to be smart.

10
Time Scales

• 0.5-5 msec for spike initiation.
• 0.1-5 msec for activation of active conductances
  in dendritic trees.
• 2-20 msec for neuron interactions.
• About three times as fast in the auditory system.
• Excitatory cells tend to be about five times as
  fast as inhibitory cells.

11
Transmission Speeds

• Roughly speaking, transmission speeds in
  myelinated axons are on the order of 10 meters a
  second. This is faster with low time constants
  (RmCm, discussed later) and large axon diameters.
• Unmyelinated axons are slower and tend to lose
  signal strength through the cell membrane.
• We usually allow about 10 msec per cell in a
  multi-layer system, but most of this involves the
  dynamics of the cell.
• In the auditory system, allow about 1 msec per
  cellits designed to be fast!

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Chemical dynamics

• The excitability of neurons depends on the flow
  of ions. These have different concentrations in
  the fluid inside and outside the neuron.
• Ions dont move far in the brain or body, so
  these flows involve small local concentration
  differences and small ionic motions.

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Ions obey physical laws

• These result in potential differences if the
  concentration of ions on the two sides of a
  membrane differ.
• The Nernst equation (1888) describes this
• Eion RT/zF x ln(Iono/Ioni)
• where Eion is the potential across the membrane
  that keeps ions from flowing. E.g,
• EK 61.5 log10(Iono/Ioni) at body
  temperature
• R is the thermodynamic gas constant, T is the
  absolute temperature, F the faraday, and z the
  valence.

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How this calculation works

• In squid, Ioni for K is about 400 mM
  (millimole), and Iono is about 20 mM, so EK is
  about -76 mV.
• In mammals, these concentrations are different,
  so EK is about -103 mV. ENa is about 62 mV. ECl
  is about -75 mV. This implies Cl- functions as a
  stabilizing but not a hyperpolarizing ion species
  in mammals. This is known as shunting
  inhibition.
• In the basal ganglia, the reversal potential of
  Cl- is actually fairly close to the action
  potential threshold.
• Ca is carefully sequestered in the neuron so
  Eca is quite positive. Spikes are generated at
  low negative Vm.

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Typical ionic concentrations

• Outside
• Na 117
• K 3
• Cl- 120
• Other- 0

• Inside
• Na 30
• K 90
• Cl- 4
• Other- 116
• These negative ions are typically bound to
  proteins and cannot flow.

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Ion channels change shape

• They contain charges, and the charges move in
  response to voltage.
• They are physical objects and respond to forces.
• They have binding sites and molecules binding
  there produce shape changes.
• These shapes gate ion flow.
• Some act as pumps, using ATP or Na ion flow to
  move other ions.

17
Hodgekin-Huxley flow
Beeman, 2005, Introduction to Realistic Neural
Modeling, http//www.wam-bamm.org/Tutorials/genes
is-intro/genesis-intro.html
18
Electrodynamics

• Each compartment is treated like a battery
• The compartments each have a membrane potential
  and are linked together via axial resistances.
• Solving these cable equations allows you to
  model these cells in some detail.

19
A generic compartment
Beeman, 2005, Introduction to Realistic Neural
Modeling, http//www.wam-bamm.org/Tutorials/genes
is-intro/genesis-intro.html
20
Definitions

• CmMembrane capacitance
• RmMembrane resistance
• EmMembrane battery (or ionic pump)
• GkVoltage dependent conductance (1/Rk)
• EkAnother battery
• IinjInjected current
• RaAxial resistance
• VmMembrane potential

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CmMembrane capacitance

• Describes energy stored by ions attracted to each
  other by the voltage difference (typically
  -70x10-3 volts) between the two sides of the
  membrane and lined up along the membrane on both
  sides.
• If the voltage difference across the membrane
  changes, the capacitance drives a current flow.
• The specific capacitance per square centimeter of
  membrane area is between 0.7 and 1x10-6 farads.

22
RmMembrane resistance

• This is the resistance in ohms times membrane
  area in cm2 to current flow through the cell
  membrane.
• The inverse of Rm is Gm, the specific leak
  conductance, measured in siemens per square
  centimeter. This can depend on the membrane
  potential and dynamic processes.
• For an ion, i, the current flow satisfies the
  following equation
• Ii(t) Gi(V(t),t) x (V(t) - Ei)
• where Ei is the reversal potential for that ion.

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EmMembrane battery (or ionic pump)

• This is the voltage of the membrane pump for a
  given ion. The units are volts.

24
GkVoltage dependent conductance (1/Rk)

• This is a conductance that depends on membrane
  potential.
• For example, the Na and Ca conductances
  involved in action potential generation and
  active dentrites activate at low negative
  membrane potentials. This causes the membrane
  potential to become positive rapidly. These
  conductances eventually close, and the K/Na
  exchange pump then takes the membrane potential
  back to negative values.

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EkAnother battery

• In the reverse direction from Em. This is a
  relatively large capacity sodium-potassium pump
  that is responsible for maintaining the resting
  membrane potential.
• This usually dominates in the resting neuron.

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IinjInjected current

• The magnitude of the injected current is used to
  solve the circuit equation.

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RaAxial resistance

• Charge flows within the neuron down the axes of
  the dendrites (and axons) and meets resistance,
  measured in ohms.
• The larger the axial resistance, the more slowly
  that ions travel from compartment to compartment,
  and the more time required for a synaptic signal
  to propagate to the soma.
• The larger the membrane conductance, the more
  charge is lost through the membrane and the less
  that actually affects the somatic membrane
  potential.

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VmMembrane potential

• Varies, which is why neurons are excitable
  cells
• Resting potentials are dominated by the potassium
  pump, and are about -40 to -100 millivolts (-50
  to -80 x 10-3 V)
• The reversal potential for sodium (the Vm at
  which there is no Na flow is slightly positive)
  is about 62 mV
• Chloride (Cl-) has a negative reversal potential
  that varies quite a bit around -75 mV
• The action potential threshold is normally around
  0 mV, but may be as negative as -40 mV.

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Various channels

• Some channels just exist and are insensitive to
  voltage. Some form batteries and pump an ion
  species in or out. Some are stretch receptors.
• Some are ligand-gated, opening or closing as a
  some chemical binds to them either on the inside
  or outside.
• Some are voltage-gated, changing configuration as
  a function of the membrane potential.
• Some are a combination, both voltage-gated and
  ligand-gated. NMDA channels have Glu as a ligand,
  but only open at certain membrane potentials that
  can eject a resident Mg ion to allow Ca flow.

30
Putting it all together

• In compartmental modelling, we literally put it
  all together, defining the topology of the model
  neuron from various dendritic compartments, the
  soma, the axon, channel types, and chemical
  synapse types.
• Cell to cell communication is slower, involving
  neurotransmitters.
• You can also have electrical synapses, where the
  ions flow directly between cells, producing
  graded potentials.

31
Modelling the axon

• Usually we model the axon as a simple delay from
  action potential (AP) generation at the soma to
  transmitter release at the synapse.
• The action potential is regenerative, so the
  magnitude is the same at all points on the axon.
  Its also not large enough to affect the state of
  the soma once generated. This is different from
  the dendritic response.
• Transmitter release involves Ca inflow due to
  the AP opening voltage-sensitive channels at the
  synapse, followed by vesicles (of fixed volume)
  docking on the membrane and quantal transmitter
  release. The number of vesicles varies.

32
Modelling the soma

• The cell body or soma is usually treated as a
  simple spherical bag. The dendrites are treated
  as cylindrical extensions of the soma.
• The voltage changes produced by the dendrites are
  treated usually as affecting the soma as a whole
  or sometimes as spreading over the membrane from
  the dendrites.
• The soma separates the dendrites topologically,
  so the dendrites communicate solely by their
  affect on the membrane potential of the soma.
• Action potentials are generated in the soma and
  propagate down the axon and back into the
  dendrites.

33
Modelling the dendrites

• The dendrites are treated as cylindrical bits of
  cell (compartments) with a membrane connecting
  the circular ends. The ends communicate to other
  dendrites, are closed off, are dead, or are
  attached to the soma.
• Branching occurs between dendritic compartments.
• Neural spines are usually ignored and synapses
  are usually given no spatial dimension.
• Synapses attach to dendritic compartments or
  directly to the soma.
• Both active and passive conductances may be
  modeled.

34
Modeling a neuron as a whole
Beeman, 2005, Introduction to Realistic Neural
Modeling, http//www.wam-bamm.org/Tutorials/genes
is-intro/genesis-intro.html
35
For the linguists

• Linguistic processes are higher level than the
  single compartmental neuron or small neural
  networks.
• Pülvermüller (Cambridge) and Schmidle (here at
  UoS) think they involve cell assemblies. These
  are large groups of cortical neurons that can
  have various states, including active, primed
  (ready to become active), depressed (unable to
  become active), and inactive or waiting. They can
  generate surprise signals when activated by
  incorrect syntax or semantics.
• The systems probably involved include the
  auditory, prefrontal, and motor cortex, and the
  basal ganglia.
• Some of my lectures will examine these systems.

36
The Hodgkin-Huxley work

• Based on outstanding experimental work with the
  squid giant neuron axons (which is responsible
  for escape responses).
• These are up to 1mm in diameter, large enough
  that recording electrodes could be inserted.
• Very resilient and function even with the
  internal contents squeezed out and replaced.

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Nature of the experiments

• Potassium, sodium, and chloride concentrations
  were controlled and the membrane potential was
  plotted.
• In live cells, the system is kept stable by the
  sodium-potassium pump.
• Action potentials involve a regenerative process
  that is triggered by a depolarisation of the cell
  membrane.

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The action potential process

• IK GKn4(V-EK)
• n describes the state of one of four proteins in
  the potassium channel (open or closed)
• INa GNa m3h(V-ENa)
• m describes the state of one of three proteins in
  the sodium channel (open or closed)
• h describes the state of another protein in the
  channel that inactivates the transfer of sodium
  ions after a while.
• There is also a leak conductance, Gm, that is
  insensitive to voltage.

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The complete equation

• CmdV/dt
• GNam3h(ENa-V) GKn4 (EK-V) Gm(Vrest-V)
  Iinj(t)
• This equation plus the equations for the three
  rate constants is the 4-dimensional HH model for
  the space-clamped axon.
• This process models action potential generation
  in some detail.

40
Conclusions

• The associated tutorial is chapter 4 of Bower and
  Beeman. You will replicate these experiments
  computationally.