Single neuron modelling

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Single neuron modelling

• Integrate and Fire
• Model
• Danny meisler
• Koby lion

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Introduction

• Most biological neurons communicate by short
  electrical pulses call action potentials or
  spikes
• In contrast to standard neuron model used
  artificial neuron networks IFN do not rely on a
  temporal average over the pulses
• In IFN the pulsed nature of the neuronal signal
  is taken into account and considered as
  potentially relevant for coding and information
  processing

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Introduction-continues

• On IFN model pulses are treaded as formal events
• This in no real drawback since in biological
  spike train ,all action potentials of the neuron
  have roughly the same form
• The time course of the action does not carry any
  information

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Introduction - continues

• IFN model is phenomenological description on an
  intermediate level of detail
• Compare to other single-Cell models they offer
  several advantages
• Moreover dynamics in networks of IFN can be
  analyzed mathematically

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Contents

• Electrical properties of neurons (membrane
  equation, Nernst equation etc)
• Simple models of Integrate and Fire
• Adding conductance
• Comparison between models
• Formal Models
• Biophysical Models

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• Neurons are enclosed by a membrane separating
  interior from extra cellular space
• The concentration of ions inside is different
  (more ve) to that in the surrounding liquid

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• -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.

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• 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.

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• The ion channels in the membrane lower the
  effective membrane resistance by a factor of
  10,000 (depending on density, type etc)

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Membrane capacitance and resistance

• 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
• k Also have ion pumps which expend energy to
  maintain the differences in concentrations inside
  and outside

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Membrane capacitance and resistance

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

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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

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Membrane capacitance and resistance

• 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
• Membrane time constant tm RmCm sets the basic
  time-scale for changes in the membrane potential
  (typically between 10 and 100ms)

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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

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Nernst equn and equilibrium potential

• 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)
• Z - is no. of extra protons of ion
• VT - is a constant (from thermal energy of ions)
• E - is equilibrium potential

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• Solve to get Nernst Equation
• From Nernst equation get equilibrium potentials
  of channelsEK 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)

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• A conductance with an equilibrium potential E
  tends to move membrane potential V towards E eg
  if V gt EK K ions will flow out of neuron and so
  hyper polarise it
• Conversely, as Na and Ca have ve Es normally V
  lt E and so ions flow in and depolarise neuron

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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
• Jj 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)

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Membrane current

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

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Membrane current

• 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

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Integrate and fire models

• These models basically assume that action
  potentials are simply spikes occurring when the
  membrane potential reaches a threshold Vth
• After firing membrane potential is reset to a
  Vreset ltVth
• Simplifies the modelling dramatically as we only
  deal with sub threshold membrane potential
  dynamics
• Can be modelled at various levels of rigour
  depending on simplifying assumptions used

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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
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Perfect integrate and fire

• Leak free capacitance
• With steady DC input, acts like Current to
  frequency converter
• Producing regular output pulses at a rate
  depending on the input current
• If input arrives in pulses, acts as divide by N
  counter.
• Its output rate depends on the average of inputs
  rates

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Leaky integrate and fire

• Will only fire if the excitatory input is strong
  enough to overcome the leak.
• tauRC
• When tau is larger than mean time between output
  spikes, the leak is insignificant.
• When tau is much smaller than the average output
  interval, then production of spike depends on
  input fluctuation.

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Leaky (Passive) integrate and fire
A passive model which assumes NO active
conductance. Therefore
Multiplying 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
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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)
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Integrate and fire neuron with Active
conductance

• To simulate non linear ion channels we add an
  active resistor.
• Their conductance depend on voltage.
• For instance, an action potential occurs when
  the membrane potential becomes depolarized enough
  that voltage controlled sodium channels open,
  initiating the fast positive feedback event of a
  spike.

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Integrate and fire neuron with Active
conductance
Spike-rate adaptation
Include an additional current in the model
The conductance relaxes exponentially to 0
Whenever the neuron fires a spike is
increased
Model with spike-rate adaptation
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Integrate-and-fireInput summation in time
Widely spaced Independent
Closely spaced Saturating. Note similarity to
charging curve
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Conceptually similar to ANN neuron
Analog inputs specified by rate
Weak input
Output rate is a function of sum of inputs.
S
Strong input
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Integrate-and-fire neuronSummation and resetting
A
B
Threshold
V
Output
time
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Biophysical neuronLogic computation
Input 1
Input 2
Sum
OR
AND
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Neuron Models Summery

• Formal models
• McCulloch-Pitts model
• Perceptron model
• Hopfield Neurons
• Biophysical models
• Integrate and Fire
• The Hodgkin Huxley Model

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FORMAL MODELS

• Easy to understand and analyze the models that
  are least like real neurons
• Each model neuron combines many inputs,
  excitatory and inhibitory , in to a single
  input
• Each neuron has at least one internal state
  variable (cells average membrane potential),
  which increases monotonically with total amount
  of excitatory input and decreases with inhibitory
  input
• Thus, the neuron is constrained to add up its
  positive and negative inputs, and cant
  independently assign an arbitrary value to each
  of its possible input combinations

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FORMAL MODELS

• McCulloch-Pitts model
• Has many progeny present in digital circuits, in
  the form of logic gates
• The explicit assumptions of this model are that
  each binary pulse represents a logical
  statement , and each neuron performs an exact,
  nose-free , synchronous computation on its input
  pulses
• If any one of the model neurons inhibitory input
  is active the output is inactive or shot off
• Otherwise, all the active excitatory input Xi are
  multiplied by their synaptic weights Wi and than
  added up

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FORMAL MODELS

• McCulloch-Pitts model
• Only if this activity level exceeds a preset
  threshold ? is output active y 1 if
  SWiXi gt 0
• 0
  otherwise
• McCulloch and Pitt showed that a large enough
  number of such units , with weights and
  connections set property ands synchronously ,
  could in principle perform any possible
  computation

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FORMAL MODELS McCulloch-Pitts model
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FORMAL MODELS

• Perceptron model
• Rosenblatts Perceptron model is formally similar
  to the McCulloch and Pitt model
• Have synchronous inputs and producing outputs
  between 0 to 1.
• But the Perceptron creates a real values (not
  binary), representing the average firing rate of
  the cell
• The internal variable V of a Perceptron is the
  weighted sum of its input V SWiXi

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FORMAL MODELS

• Perceptron model
• A threshold or bios ? is subtracted from V and is
  then passed through a continuous and
  monotonically increasing function g
• 
  Y g (V ?)
• The nonlinear function g is sigmoidal

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FORMAL MODELSPerceptron model
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FORMAL MODELS

• Hopfield Neurons
• In Hopfields binary model , the output of neuron
  i is the step function of Vi and the threshold ?
  
• 
  Yi 0 if Vi lt ?
• 
  1 if Vi gt ?
• Unlike the McCulloch Pitt or Perceptron model,
  each Hopfields neuron update its state at a
  random time, independently of any other neurons
• Similar to Perceptron in isolation, but can act
  associative memories in highly interconnected
  network

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FORMAL MODELSHopfield Neurons
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BIOPHYSICAL MODEL

• Although many crucial properties of real neuron
  remain unknown, biophysical model incorporate
  some known properties of neural tissue
• Like real, spiking neurons these models produce
  spikes rather than continuous valued outputs
• Integrate and Fire Model
• The Hodgkin Huxley Model

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BIOPHYSICAL MODEL

• Integrate and Fire Model
• Divides membrane behavior conceptually into two
  regimes.
• A prolonged period of linear integration
• A sudden firing
• Relaxes the requirements that a single set of
  continuous differential equations describe the
  cells two very different regimes.
• Easily implement using simple electronic circuits.

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BIOPHYSICAL MODEL Integrate and Fire Model
Vth
time
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BIOPHYSICAL MODEL

• The Hodgkin Huxley Model
  
• Biophysically much more accurate single cell
  models that can account for the very complex, non
  stationary behavior of real neurons
• The dynamics are modeled by numerous coupled
  ,nonlinear differential equations that describe
  the behavior of continuous currents that depend
  in a nonlinear manner on the membrane potential

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BIOPHYSICAL MODEL

• The Hodgkin Huxley Model
  
• Although this model is powerful, if suffers from
  the drawback that it requires detail knowledge of
  a myriad of parameters. It is frequently
  difficult to properly constrain all these degrees
  of freedom.

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BIOPHYSICAL MODEL The Hodgkin Huxley Model
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• THE
• END