1Integrating neural models using exact integration
2=================================================
3
4The simple integrate-and fire model
5~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
6
7For the simple integrate-and-fire model the voltage :math:`V` is given as a solution of the equation:
8
9.. math::
10    C\frac{dV}{dt}=I.
11
12This is just the derivate of the law of capacitance :math:`Q=CV`. When an input current is applied, the membrane voltage increases with time until it reaches a constant threshold :math:`V_{\text{th}}`, at which point a delta function spike occurs.
13
14A shortcoming of the simple integrate-and-fire model is that it implements no time-dependent memory. If the model receives a below-threshold signal at some time, it will retain that voltage boost until it fires again. This characteristic is not in line with observed neuronal behavior.
15
16The leaky integrate-and fire model
17~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
18
19In the leaky integrate-and-fire model, the memory problem is solved by adding a "leak" term :math:`\frac{-1}{R}V` (:math:`R` is the resistance and :math:`\tau=RC`) to the membrane potential:
20
21.. math::
22    \frac{dV}{dt}=\frac{-1}{\tau}V+\frac{1}{C}I.
23    :label: membrane
24
25This reflects the diffusion of ions that occurs through the membrane when some equilibrium is not reached in the cell.
26
27Solving a  homogeneous linear differential equation
28~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
29
30To solve :math:numref:`membrane` we start by looking at a simpler differential equation:
31
32.. math::
33    \frac{df}{dt}=af\text{, where } f:\mathbb{R}\to\mathbb{R} \text{ and } a\in\mathbb{R}.
34
35Here the solution is given by :math:`f(t)=e^{at}`.
36
37Solving a non-homogeneous linear differential equation
38~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
39When you add another function :math:`g` to the right hand side of our linear differential equation,
40
41.. math::
42    \frac{df}{dt}=af+g
43
44this is now a non-homogeneous differential equation. Things (can) become more complicated.
45
46Solving it with variation of constants
47^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
48
49This kind of differential equation is usually solved with "variation of constants" which gives us the following solution:
50
51.. math::
52    f(t)=e^{ct}\int_{0}^t g(s)e^{-cs}ds.
53
54This is obviously not a particularly handy solution since calculating the integral in every step is very costly.
55
56Solving it with exact integration
57^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
58
59With exact integration, these costly computations can be avoided.
60
61Restrictions to :math:`g`
62------------------------
63But only for certain functions :math:`g`! I.e. if :math:`g` satisfies (is a solution of):
64
65.. math::
66    \left(\frac{d}{dt}\right)^n g= \sum_{i=1}^{n}a_i\left(\frac{d}{dt}\right)^{i-1} g
67
68for some :math:`n\in \mathbb{N}` and a sequence :math:`(a_i)_{i\in\mathbb{N}}\subset \mathbb{R}`.
69
70For example this would be the case for :math:`g=\frac{e}{\tau_{syn}}t e^{-t/\tau_{\text{syn}}}` (an alpha funciton), where :math:`\tau_{\text{syn}}` is the rise time.
71
72Reformulating the problem
73^^^^^^^^^^^^^^^^^^^^^^^^^
74
75The non-homogeneous differential equation is reformulated as a multidimensional homogeneous linear differential equation:
76
77.. math::
78    \frac{d}{dt}y=Ay
79
80where
81
82.. math::
83    A=\begin{pmatrix}
84        a_{n}  & a_{n-1} & \cdots & \cdots & a_1    & 0 \\
85        1      & 0       & \cdots & 0      & 0      & 0 \\
86        0      & \ddots  & \ddots & \vdots & \vdots & \vdots \\
87        \vdots & \ddots  & \ddots & 0      & 0      & 0 \\
88        0      & 0       & \ddots & 1      & 0      & 0 \\
89        0      & 0       & \cdots & 0      & \frac{1}{C} & -\frac{1}{\tau} \\
90    \end{pmatrix}
91
92by choosing :math:`y_1,...,y_n` canonically as:
93
94.. math::
95    \begin{align*}
96        y_1 &= \left(\frac{d}{dt}\right)^{n-1}g\\
97        \vdots &= \vdots\\
98        y_{n-1} &= \frac{d}{dt}g\\
99        y_{n} &= g\\
100        y_{n+1} &= f.
101    \end{align*}
102
103This makes ist very easy to determine the solution as
104
105.. math::
106    y(t)= e^{At}y_0
107
108and
109
110.. math::
111    y_{t+h}=y(t+h)=e^{A(t+h)}\cdot y_0=e^{Ah}\cdot e^{At}\cdot y_0=e^{Ah}\cdot y_t.
112
113This means that once we have calculated :math:`A`, propagation consists of multiplications only.
114
115Example: The leaky integrate and fire model with alpha-function shaped inputs (iaf_psc_alpha)
116^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
117
118The dynamics of the membrane potential :math:`V` is given by:
119
120.. math::
121    \frac{dV}{dt}=\frac{-1}{\tau}V+\frac{1}{C}I
122
123where :math:`\tau` is the membrane time constant and :math:`C` is the capacitance. :math:`I` is the sum of the synaptic currents and any external input:
124
125Postsynaptic currents are alpha-shaped, i.e. the time course of the synaptic current :math:`\iota` due to one incoming spike is
126
127.. math::
128    \iota (t)= \frac{e}{\tau_{syn}}t e^{-t/\tau_{\text{syn}}}.
129
130The total input :math:`I` to the neuron at a certain time :math:`t` is the sum of all incoming spikes at all grid points in time :math:`t_i\le t` plus an additional piecewise constant external input :math:`I_{\text{ext}}`:
131
132.. math::
133    I(t)=\sum_{i\in\mathbb{N}, t_i\le t }\sum_{k\in S_{t_i}}\hat{\iota}_k \frac{e}{\tau_{\text{syn}}}(t-t_i) e^{-(t-t_i)/\tau_{\text{syn}}}+I_{\text{ext}}
134
135:math:`S_t` is the set of indices that deliver a spike to the neuron at time :math:`t`, :math:`\tau_{\text{syn}}` is the rise time and :math:`\iota_k` represents the "weight" of synapse :math:`k`.
136
137Exact integration for the iaf_psc_alpha model
138---------------------------------------------
139
140First we make the substitutions:
141
142.. math::
143    \begin{align*}
144        y_1 &= \frac{d}{dt}\iota+\frac{1}{\tau_{syn}}\iota \\
145        y_2 &= \iota \\
146        y_3 &= V
147    \end{align*}
148
149for the equation
150
151.. math::
152    \frac{dV}{dt}=\frac{-1}{Tau}V+\frac{1}{C}\iota
153
154we get the homogeneous differential equation (for :math:`y=(y_1,y_2,y_3)^t`)
155
156.. math::
157    \frac{d}{dt}y= Ay=
158    \begin{pmatrix}
159    \frac{1}{\tau_{syn}}& 0 & 0\\
160    1 & \frac{1}{\tau_{syn}} & 0\\
161    0 & \frac{1}{C} & -\frac {1}{\tau}
162    \end{pmatrix}
163    y.
164
165The solution of this differential equation is given by :math:`y(t)=e^{At}y(0)` and can be solved stepwise for a fixed time step :math:`h`:
166
167.. math::
168    y_{t+h}=y(t+h)=e^{A(t+h)}y(0)=e^{Ah}e^{At}y(0)=e^{Ah}y(t)=e^{Ah}y_t.
169
170The complete update for the neuron can be written as
171
172.. math::
173    y_{t+h}=e^{Ah}y_t + x_{t+h}
174
175where
176
177.. math::
178    x_{t+h}+\begin{pmatrix}\frac{e}{\tau_{\text{syn}}}\\0\\0\end{pmatrix}\sum_{k\in S_{t+h}}\hat{\iota}_k
179
180as the linearity of the system permits the initial conditions for all spikes arriving at a given grid point to be lumped together in the term :math:`x_{t+h}`. :math:`S_{t+h}` is the set of indices :math:`k\in 1,....,K` of synapses that deliver a spike to the neuron at time :math:`t+h`.
181
182The matrix :math:`e^{Ah}` in the C++ implementation of the model in NEST is constructed `here <https://github.com/nest/nest-simulator/blob/b3fc263e073f46f0732c10efb34fcc90f3b6771c/models/iaf_psc_alpha.cpp#L243>`_.
183
184Every matrix entry is calculated twice. For inhibitory postsynaptic inputs (with a time constant :math:`\tau_{syn_{in}}`) and excitatory postsynaptic inputs (with a time constant :math:`\tau_{syn_{ex}}`).
185
186And the update is performed `here <https://github.com/nest/nest-simulator/blob/b3fc263e073f46f0732c10efb34fcc90f3b6771c/models/iaf_psc_alpha.cpp#L305>`_. The first multiplication evolves the external input. The others are the multiplication of the matrix :math:`e^{Ah}` with :math:`y`. (For inhibitory and excitatory inputs)
187
188References
189~~~~~~~~~~
190
191.. [1] RotterV S & Diesmann M (1999) Exact simulation of time-invariant linear
192    systems with applications to neuronal modeling. Biologial Cybernetics
193    81:381-402. DOI: https://doi.org/10.1007/s004220050570
194