DOING PHYSICS WITH MATLAB

 

SPIKING NEURONS

LEAKY INTEGRATE-AND-FIRE MODEL

 

Ian Cooper

     Any comments, suggestions or corrections, please email me at

    matlabvisualphysics@gmail.com

 

 

MATLAB

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ns_LIF_002.m

LIF model of the time evolution of the membrane potential due to an external stimulus

 

 

 

NEURONS

Body fluids are good electrical conductors because salts and other molecules dissociate into positive and negative ions. The inside of an axon is filled with an ionic fluid that is separated from the surrounding body fluid by a thin membrane that is from about 5 nm to 10 nm thick. The ionic solutes in the extracellular fluid are mainly Na+ and Cl- ions. In the intracellular fluid, the positive ions are mainly K+ and the negative ions are mainly large negatively charged organic ions.  Hence, there is a large concentration of Na+ ions outside the axon and a large concentration of K+ ions inside the axon. The concentration of the different ion species does not equalize by diffusion because of the special properties of the cell membrane. In the resting state when the axon is non-conducting, the axon membrane is highly permeable to K+ ions, slightly permeable to Na+ ions and impermeable to large negative organic ions. More K+ ions leak out of the cell than Na+ ions that leak into the cell. This leaves the inside of the cell more negative than the outside. A potential difference therefore exists across the cell membrane because of the difference in the concentration of ions in the extracellular and intracellular fluids. This potential difference is called the membrane potential vm(t). The outside of the cell is taken as the reference potential 0 V. The resting membrane potential has a strong negative polarization and is constant at about -65 mV. This negative membrane potential restricts the further diffusion of the K+ to the outside of the cell so that equilibrium is established where the electrical forces balances the chemical forces. Thus, the membrane acts as a capacitor in parallel with a resistor.

 

The mechanism for the generation of an electrical signal by a neuron is conceptually simple. When a neuron receives a sufficient stimulus from another neuron, the permeability of the cell membrane changes. As a result of the changes in membrane permeability, the sodium ions first rush into the cell while the potassium ions flow out of it. The movement of the ions across the membrane constitutes an electric current signal which propagates along the axon to its terminations. These membrane currents depolarize the cell so that the interior of the cell becomes positive and a neuronal voltage signals is generated. These short voltage pulses are called spikes or action potentials and have a duration of less than a few milliseconds and have a peak about +40 mV. The action potential propagates along an axon without a change in shape.

 

LEAKY INTEGRATE-AND-FIRE NEURON MODEL

We can start the analysis of the electrical properties of a neuron using the simplest possible model to generate action potentials, known as the leaky integrate-and-fire (LIF) model. The membrane of a nerve cell separates the intracellular and extracellular fluids with inside of the cell more negative than the outside of the cell in its resting state. The electrical properties of the cell membrane are modelled as a parallel circuit consisting of the membrane capacitance C and the membrane resistance R in series with a battery with an emf equal to resting potential vrest and driven by some external stimulus (figure A).  

          Fig. A.   RC circuit model of the nerve cell membrane used in the LIF model.

 

Capacitor current 

        

Leakage current through resistor

        

Kirchhoff’s current law

     

 

       (1)      

 

Equation 1 is the leaky integrate-and-fire (LIF) differential equation for the membrane potential  where is the membrane time constant and  is the resting potential of the membrane  .

 

We can solve equation 1 using the finite difference method to compute the membrane potential at a series of time steps of duration

          (2)      

 

The spiking events are not explicitly modelled in the LIF model. Instead, when the membrane

potential vm(t) reaches a certain threshold vTH (spiking threshold), it is instantaneously reset to a lower value vreset (reset potential) and the leaky integration process described by equation 1 continues with the membrane potential set at vreset. However, we can artificially produce a spike when  by setting   then .

 

To add just a little bit of realism to the dynamics of the LIF model, it is possible to add an absolute refractory period  immediately after a spike is generated when .  During the absolute refractory period, vm can be clamped to vreset and the leaky integration process re-initiated following a delay of  after the spike.

 

 

SIMULATIONS

 

The mscript  ns_LIF_002.m  can be used to solve equation 2 for different time dependent external stimuli.

 

Typical parameters used in the modelling are:

                    

                                    

The variable  flagS  is used to select the function for the external stimulus current input and  flagF  for the calculation of the firing rate and f – I curve.

 

 

Simulation 1:     Subthreshold regime /  Free solutions         

Exponential decay of membrane potential to resting potential. The larger the time constant, then the more slowly the membrane potential decreases towards the resting value for the membrane potential.

 

Fig. 1A.   Depolarization of the membrane. (flagS = 1) 

Fig. 1B.  Depolarization of the membrane.  (flagS = 1)      

Fig. 1A.   Hyperpolarization of the membrane. (flagS = 1) 

Fig. 1B.  Hyperpolarization of the membrane.  (flagS = 1)      

 

 

Simulation 2:   Subthreshold regime (pulse input)

 

Fig. 2.      A series of input pulses results in a linear summation of the membrane response to each pulse.  The membrane potential remains at a value less than the threshold potential. No spikes are generated.    (flagS = 2)

 

 

Simulation 3:      Subthreshold regime (pulse input)

When a short pulse acts as the external stimulus most of the charge Q is deposited onto the capacitor and very little charge passes through the resistor. Initially the capacitor is charged and then discharges through the resistor as the input stimulus value goes to zero.

 

Let the width of the input pulse be  where    and the charge then from equation 2, we get

      

        (3)        

 

In the subthreshold regime, equation 3 implies that for short input pulses which have the same area , the membrane potential peak values will be the same as illustrated in figure 3 where in both examples Q = 20 nC (the areas under the two current vs time graphs).

     Fig. 3.     Membrane potential response to two pulse inputs. (flagS = 4)

 

 

Simulation 4:        Spiking neuron

Fig. 4.      An action potential is produced when the membrane potential reaches its threshold value. After the neuron has fired, the membrane potential is reset to the reset voltage.  If Iext > 1.2 nA a spike is generated. (flagS = 4).

 

 

Simulation 5:     Spiking neuron with a step input

A step input stimulus results in a continual firing of the neuron at regular intervals. In figure 5a, the absolute refractory period is set to zero, whereas in figure 5b, the refractory period is .  (flagS = 3).

 

Fig. 5a.   Time evolution of the membrane potential and the external current input stimulus for a zero absolute refractory period.  .  

Fig. 5b.   Time evolution of the membrane potential and the external current input stimulus for a non-zero absolute refractory period.

 

 

Simulation 6        A ramp input stimulus produces action potentials with an increasing firing rate as the input strength increases (figure 6   flagS = 5)

 

Fig. 6a.   Firing rate of neuron increases as strength of the input stimulus increases. The absolute refractory period is set to zero. 

 

Matlab Command Window

Interspike times   ISI   [ms] 

 14.48  10.90  9.14  8.04  7.28  6.68  6.24  5.86

  

Neuron firing rate  [Hz]  

 69.05  91.72  109.39  124.35  137.34  149.67  160.22  170.61  

mean firing rate  f =  126.54

 

Fig. 6b.   Firing rate of neuron increases as strength of the input stimulus increases. A non-zero absolute refractory period reduces the firing rate.   

 

Matlab Command Window

Interspike times   ISI   [ms] 

 18.02  14.46  12.72  11.64  10.88   

Neuron firing rate  [Hz]  

 55.48  69.14  78.60  85.89  91.89  

mean firing rate  f =  76.20

 

 

Fig. 6c.     f – I  curve for a LIF neuron.

 

                         blue curve   (figure 7a)

 

                       red curve   (figure 7b)

 

An action potential is not produced until the external current exceeds a critical value which has a value of about 1.5 nA.

 

 

Simulation 7     Noisy input  

Fig. 7.     The membrane potential response to a noisy input current stimulus. At each time step the external current is randomly assigned between 0 to 2.0 nA.  The spike times are irregular.  (flagS = 6)

 

 

Simulation 8:    Synaptic current inputs

 

Consider a more realistic situation where the neuron is stimulated by pre-synaptic spikes arriving at its synapses. The pre-synaptic spikes are linearly summed to give the input current and when the threshold voltage is reached, a spike is generated.

 

Fig. 8.  Each pulse can be considered as an input from a set of pre-synaptic junctions.

 

 

 

 

 

REFERENCES

 

Emin Orhan: The Leaky Integrate-and-fire Neuron model

Wulfram Gerstner, Werner M. Kistler, Richard Naud and Liam Paninski:

Neuronal Dynamics - From single neurons to networks and models of cognition

Limitations of the Leaky Integrate-and-Fire Model