Outline: Introduction (neurons and models) Integrate and fire based neuron model Leaky integrate and fire based neuron model Spike-Response Model Mathematical Formulation Simulating Refractoriness Fitting to Experimental Data Variations of SRM Effects not captured by SRM Adaptive Exponential Integrate-and-Fire Model Definition Adaptation, Delayed spiking, Voltage Response, Initial bursting Fitting to real Neurons’ data

1. 1
ZUBIN BHUYAN (CSI 11014)
NAYANTARA KOTOKY(CSI 11025)
NIRUPAM CHOUDHURY(CSI 11033)
Basic Neuro-Engineering

2. Outline
2
 Introduction(neurons and models)
 Integrate and fire based neuron model
 Leaky integrate and fire based neuron model
 Spike-Response Model
 Mathematical Formulation
 Simulating Refractoriness
 Fitting to Experimental Data
 Variations of SRM
 Effects not captured by SRM
 Adaptive Exponential Integrate-and-Fire Model
 Definition
 Adaptation, Delayed spiking, Voltage Response, Initial bursting
 Fitting to real Neurons’ data

3. Review of the neuron
 Action potential- very rapid
change in membrane
potential when a nerve cell
membrane is stimulated.
 Resting potential (typically -
70 mV) to some positive
value (typically about +30
mV).
 Threshold stimulus &
threshold potential(generally
5 - 15 mV less negative than
the resting potential)

4. Neuron model
 Biological neuron model- mathematical
description of the properties of nerve cells.
 Artificial neuron model- aims for computational
effectiveness.

5. Artificial neuron abstraction
Consists of-
 an input with some synaptic
weight vector
 an activation
function or transfer
function inside the neuron
determining output.
 Oj=f( ∑wijei )

6. Biological abstraction
In the case of modelling a biological neuron-
 Physical analogues are used in place of abstractions
such as “weight” and “transfer function’’.
 Ion current through the cell membrane is described by
a physical time-dependent current I(t)
 Insulating cell membrane determines a capacitance
C m.
 A neuron responds to such a signal with a change
in voltage, or an electrical potential energy difference
between the cell and its surroundings, sometimes
resulting in a voltage spike called an action potential.

7. 7
Integrate-and-Fire based Neuron
Model
L. F. Abbott*, 21 May 1999, Lapicque’s introduction of
the integrate-and-fire model neuron (1907)

8. IAF model
 One of the earliest models of a neuron.
 First investigated in 1907 by Louis
Lapicque.
 Lapicque modeled the neuron using an
electric circuit consisting of a parallel capacitor
and resistor.
 When the membrane capacitor was charged to
certain threshold potential
 an action potential would be generated
 the capacitor would discharge

9. Theoretical idea
 In a biologically realistic neural network, it often
takes multiple input signals in order for a neuron to
propagate a signal.
 Multiple input signals goes from one neuron to the
next, increasing the effect of one firing by however
many connection there are(done by adjusting the
weights between each neuron).
 Every neuron has a certain threshold at which it
goes from stable to firing.
 When a cell reaches its threshold and fires, its
signal is passed onto the next neuron, which may
or may not cause it to fire.

10. contd…
 If the neuron does not fire, its potential will be raised
so that if it receives another input signals within a
certain time frame, it will be more likely to fire.
 If the neuron does fire, then the signal will be
propagated onto the next neuron.
 After this, the just-fired neuron goes into a refractory
state, in which it doesn't respond to or propagate
input signals from other neurons.
 This increased potential to fire starts to dampen
soon after the input is received.

11. Mathematical representation
 A neuron is represented in time by
 When an input current is applied, the
membrane voltage increases with time until it
reaches a constant threshold Vth , at which point
a delta function spike occurs and the voltage is reset
to its resting potential, after which the model
continues to run.
 The firing frequency of the model increases linearly
without bound as input current increases.

12. contd…
 By introducing a refractory period tref , we limit the
firing frequency of a neuron by preventing it from
firing during that period.
 Firing frequency as a function of a constant
input current is:
 Shortcoming:
 It implements no time-dependent memory. If the
model receives a below-threshold signal at some
time, it will retain that voltage boost forever until it
fires again.

13. 13 Leaky integrate and fire model
http://lcn.epfl.ch/~gerstner/SPNM/node26.html

14. Description
 In the leaky integrate-and-fire model, the
memory problem is solved by adding a "leak"
term to the membrane potential.
 It reflects the diffusion of ions that occurs
through the membrane when some equilibrium
is not reached in the cell.
(1)
 Rm is the membrane resistance
 threshold Ith = Vth / Rm

15. contd…
 When input current exceeds threshold Ith , it
causes the cell to fire, else it will simply leak
out any change in potential.
 firing frequency is:

16. contd…
 We multiply equation
(1) by R(resistance)
and considering Ω=R
Cm of the “leaky
integrator” to get-

17. 17 Spike-Response Model
Izhikevich, E.M. (2001), Resonate-and-fire neurons,
•
Neural Networks, 14:883-894
Izhikevich E.M. (2003), Simple model of spiking
•
neurons, IEEE Transactions On Neural Networks,
14:1569-1572

18. Spike-response model..
18
 Generalization of the leaky integrate-and-
fire model
 Gives a simple description of action
potential generation in neurons
 Spike response model includes refractoriness

19. SRM: Mathematical Formulation
19
 The membrane potential in the spike response model is
given by
 Here t’ is the firing time of the last spike
 η describes the form of the action potential
 Κ the linear response to an input pulse
 I(t) is a stimulating current
 The next spike occurs if the membrane potential u hits a
threshold Ɵ(t-t’) from below in which case t’ is updated

20. SRM: Mathematical Formulation (..contd)
20
 The threshold Ɵ is not fixed but depends on
the time since the last spike
 threshold is higher immediately after a spike
 then it decays back to its resting value
 The spike shape η is a function of the time
since the last spike
 Itcan describe a depolarizing, hyperpolarizing, or
resonating spike-after potential

21. SRM: Mathematical Formulation (..contd)
21
 The responsiveness Κ to an input pulse
depends on the time since the last spike
 since many ion channels are open
 typically the effective membrane time constant after a
spike is shorter
 The time course of the response Κ can include
 a single exponential
 combinations of exponentials with different time
constants
 or resonating behavior in form of a delayed oscillation
 (This is the case if the standard Hodgkin-Huxley model is
approximated by the Spike Response Model)

22. SRM: Simulating Refractoriness
22
 Refractoriness can be modelled as a
combination of
 increased threshold
 hyperpolarizing afterpotential
 and reduced responsiveness after a spike
 [as observed in real neurons (Badel et al., 2008)]

23. SRM: η (..contd)
23
 Example of a
spike
shape η with
rapid reset,
followed by a
hyperpolarizin
g action
potential,
extracted
from data

24. SRM: η (..contd)
24
 Example of a
spike shape
η with
depolarizing
afterpotential,
extracted from
data

25. SRM: Fits to experimental data
25
 SRM can be fitted to experimental data where
a neuron is stimulated

26. SRM: Fits to experimental data (…contd.)
26
 SRM fits experimental data to a high degree of
accuracy
 Predicts a large fraction of spikes with a
precision of +/-2ms, (Jolivet et al., 2006).

27. Variations of SRM: SRM0
27
 A simplified version of the SRM.
 Does not include a dependence of the response
kernel K upon the time since the last spike
 The threshold can be dynamic as before
 Easier to fit to experimental data than the full
SRM
 since it needs less data (Jolivet et al. 2006)

28. Cumulative Spike Response Model
28
 Refractoriness and adaptation are modeled by
the combined effects of the spike
afterpotentials of several previous spikes
 And not the most recent spike
 The advantage of the cumulative model is that it
accounts for adaptation and bursting

29. SRM with a cumulative dynamic
threshold
29
 Value of the threshold depends on all previous
spikes
 and not only the most recent one
 The threshold Ɵ is calculated as
tk denotes previous moments of spike firing
 Ɵo is the value of the threshold at rest
 v(t-tk) describes the effect of a spike at time tk

30. Noise in the SRM
30
 Noise can be included into the SRM by
replacing the strict threshold criterion,
u(t) = Ɵ, by a stochastic process
 The probability P of firing a spike within a very
short time Δt is
 P = ρ(t) Δt
 where the instantaneous firing rate ρ(t) is a function of
the momentary difference between the membrane
potential u(t) and the threshold Ɵ(t),
 ρ = f(u - Ɵ)

31. Effects not captured by a SRM
31
 Pharmacological blocking of ion channels
 Biophysics of the neuronal membrane is not
described explicitly
 combined effects of several ion channel are
captured
 model cannot make predictions about blocking of
individual ion channels
 Delayed spike initiation due to different
amplitude of the input pulse
 because of the strict threshold criterion
 Dependence of the threshold upon the input

32. Adaptive Exponential Integrate-and-
32
Fire Model
Brette R. and Gerstner W. (2005), Adaptive
Exponential Integrate-and-Fire Model as an
Effective Description of Neuronal Activity, J.
Neurophysiol. 94: 3637 - 3642.

33. AdEx
33
 A spiking neuron model with two equations
 The first equation describes the dynamics of
the membrane potential
 includes an activation term with an exponential
voltage dependence
 Voltage is coupled to a second equation which
describes adaptation
 Both variables are reset if an action potential has
been triggered

34. AdEx: Mathematical Definition
(…contd.)
34
 The model is described by two differential
equations
V is the membrane potential C the membrane capacitance
I the input current EL the leak reversal potential
gL the leak conductance ΔT the slope factor
VT the threshold w the adaptation variable
a is the adaptation coupling parameter τw is the adaptation time constant

35. AdEx (…contd.)
35
 Exponential nonlinearity describes the process of spike
generation and the upswing of the action potential
 a spike is said to occur at the time tf when the
membrane potential V diverges towards infinity
 The downswing of the action potential is a reset
of the voltage to a fixed Vr
at t=tf reset V→Vr
 Also, change the adaption value by b:
w = w +b

36. AdEx: Adaptation
36
 Adaptation and
regular firing of
the AdEx model
in response to a
current step;
voltage (top) and
adaptation
variable (bottom)
[Naud, Marcille, Colpath, Gerstner (2008), Firing patterns in the adaptive exponential integrate-and-fire model]

37. AdEx: Voltage Response
37
 Voltage
response of the
AdEx model to
a series of
regularly spaced
(10 Hz) current
pulse
[Naud, Marcille, Colpath, Gerstner (2008), Firing patterns in the adaptive exponential integrate-and-fire model]

38. AdEx: Initial bursting as response
38
 Voltage (X-axis) and adaptation variable  Voltage as a function of
 Resting potential marked by cross time
 reset values marked by squares
[Naud, Marcille, Colpath, Gerstner (2008), Firing patterns in the adaptive exponential integrate-and-fire model]

39. AdEx: Bursting
39
 Bursting with 3 spikes per burst in the AdEx model
 Bursting occurs when the reset value Vr is high, so that spikes are produced
quickly after reset, until adaptation builds up
[Naud, Marcille, Colpath, Gerstner (2008), Firing patterns in the adaptive exponential integrate-and-fire model]

40. AdEx: Firing Pattern
40
 Red- regular firing
 Yellow- Adaptive
 Initial bursting-
Green
 Blue- Regular
bursting
 Black- Irregular-
chaotic
Figure shows how the choice of reset of voltage (horizontal) and adaptation (vertical)
influences firing patterns

41. Fitting to real Neurons’ data
41
 The parameters of the AdEx model can be fit
to match the response of neurons
 usingsimple electrophysiological protocols
(current pulses, steps and ramps)
 AdEx model can reproduce up to 96% of the
spike times of a regular-spiking Hodgkin–
Huxley-type model
 [Brette and Gerstner, 2005]

42. Fitting to real neurons (…contd.)
42
With the same
set of
parameters,
AdEx reproduces
spikes of the
Hodgkin-Huxley
model for various
firing rates.

43. Some Limitations
43
 Single-compartment model
 But works fine
 Sodium channel activation is instantaneous
 In H-H model activation of the sodium current (via
the m variable) is rapid, but lags the evolution of the
voltage by a short time in the millisecond range
 Downswing of action potential is by resetting to
a fixed value after the spike
 Rapid potassium currents (and also partially by
sodium channel inactivation) is neglected

44. Some Limitations (…contd.)
44
 Refractoriness is only represented by the reset of
voltage and adaptation variables
 in real neurons refractoriness is due to increase in
the firing threshold and conductance after a spike and
a change in the momentary equilibrium potential
 Conductance effects are ignored, because the
adaptation variable enters as a current

45. REFERENCES
45
1. Abbott, L.F. (1999). "Lapique's introduction of the integrate-and-fire model neuron
(1907)“
2. Izhikevich, E.M. (2001), Resonate-and-fire neurons, Neural Networks, 14:883-894
3. Izhikevich E.M. (2003), Simple model of spiking neurons, IEEE Transactions On
Neural Networks, 14:1569-1572
4. Brette R. and Gerstner W. (2005), Adaptive Exponential Integrate-and-Fire Model
as an Effective Description of Neuronal Activity, J. Neurophysiol. 94: 3637 - 3642.
5. Naud, Marcille, Colpath, Gerstner (2008), Firing patterns in the adaptive
exponential integrate-and-fire model, Biological Cybernetics, DOI
10.1007/s00422-008-0264-7
6. Benda J, Herz A.V.M. (2003), A universal model for spike-frequency adaptation.
Neural Comput. 15:2523-2564.
7. http://www.scholarpedia.org/Spike-response_model (doi:10.4249/scholarpedia.1343)
8. Koch, Christof; Idan Segev (1998). Methods in Neuronal Modeling (2 ed.).
Cambridge, MA: Massachusetts Institute of Technology. ISBN 0-262-11231-0
9. http://lcn.epfl.ch/~gerstner/SPNM/node26.html

46. Thank You!
46

47. 47

48. AdEx: Delayed spiking
48
 Delayed spiking as
response of the AdEx
model to a current step
Voltage as a function of time
[Naud, Marcille, Colpath, Gerstner (2008), Firing patterns in the adaptive exponential integrate-and-fire model]