2.21. Reporting

ANNarchy includes an utility allowing to automatically generate a report based on the current structure of the network:

report(filename="model_description.tex")
report(filename="model_description.md")

If the filename ends with .tex, the LaTeX report will be generated based on the specifications provided in:

> Nordlie E, Gewaltig M-O, Plesser HE (2009). Towards Reproducible Descriptions of Neuronal Network Models. PLoS Comput Biol 5(8): e1000456.

If the filename ends with .md, the report will be generated in Markdown, so it can later be exported to pdf or html using pandoc.

report() accepts several arguments:

  • filename: name of the file where the report will be written (default: ”./report.tex”)
  • standalone: tells if the generated TeX file should be directly compilable or only includable. Ignored in Markdown.
  • gather_subprojections: if a projection between two populations has been implemented as a multiple of projections between sub-populations, this flag allows to group them in the summary (default: False).
  • title: title of the document (Markdown only)
  • author: author of the document (Markdown only)
  • date: date of the document (Markdown only)
  • net_id: id of the network to be used for reporting (default: 0, everything that was declared)

2.21.1. Content of the TeX file

report() produces a .tex file (by default report.tex in the current directory, but this can be changed by passing the filename argument) which can be directly compiled with pdflatex or integrated into a larger file:

pdflatex model_description.tex

This report consists of different tables describing several aspects of the model:

  1. Summary: A summary of the network, with a list of populations, neuron and synapse models, topologies, etc. This section may have to be adapted, as for example, ANNarchy does not make a distinction between synapse and plasticity models.
  2. Populations: A list of populations, with their respective neural models and geometries.
  3. Projections: A list of projections, with the pre- and post-synaptic populations, the target, the synapse model if any, and a description of the connection pattern.
  4. Neuron models: For each neuron model, a description of its dynamics with equations parsed using SymPy and translated to the LaTeX mathematical language.
  5. Synapse models: For each synapse model, a description of its dynamics if any.
  6. Parameters: The initial value (before the call to compile()) of the parameters of each population and projection (if any).
  7. Input: Inputs set to the network (has to be filled manually).
  8. Measurements: Measurements done in the network (has to be filled manually).

2.21.2. Content of the Markdown file

The generated Mardown file is globally similar to the LaTeX one, with additional information that make it more useful for debugging (locality of attributes, type...). The Markown file is readable by design, but it can be translated to many markup languages (html, epub, latex, pdf...) using pandoc.

To obtain a pdf from the Markdown file (supposing you have a LaTeX distribution available), just type:

pandoc model_description.md -sN -V geometry:margin=1in -o model_description.pdf

The -V argument tells LaTex to use the full page instead of the default booklet format.

To obtain a html file, use:

pandoc model_description.md -sSN --mathjax -o model_description.html

You can omit the -S option if you only want to include the code into a webpage, otherwise it is a standalone file. --mathjax is needed to display mathematical equations using the javascript library MathJax.

By default, the html file has no styling, and tables can be very ugly. With a simple css file like this one, the html page looks nicer (feel free to edit):

pandoc model_description.md -sSN --mathjax --css=simple.css -o model_description.html

If you upload your model to a github-like service (bitbucket, gitlab, gogs...), it could be a good idea to generate the README.md directly with report(). Do not forget to set a title+author+date then.

2.21.3. Documenting the network

The report is generated based entirely on the Python script. For it to make sense, the user has to provide the necessary information while defining the network:

  1. Populations must be assigned a unique name. If no name is given, generic names such as pop0 or pop1 will be used. If two populations have the same name, the connectivity will be unreadable:

    pop1 = Population(geometry=(100, 100), neuron=Izhikevich, name="Excitatory")
    pop2 = Population(geometry=(20, 20), neuron=Izhikevich, name="Inhibitory")
    
  2. User-defined neuron and synapse models should be assigned a name and description. The name should be relatively short and generic (e.g. “Izhikevich”, “BCM learning rule”), while the description should be more specific. They can contain LaTeX code, but remember to double the \ which is the escape symbol in Python strings:

LIF = Neuron(
    parameters = """
        tau = 10.0
    """,
    equations = """
        tau * dv/dt + v = g_exc
    """,
    spike = "v > 30.0",
    reset = "v = 0.0"
    name = "LIF",
    description = "Leaky Integrate-and-Fire spiking neuron with time constant $\\tau$."
)

Oja = Synapse(
    parameters = """
        eta = 10.0
        tau = 10.0 : postsynaptic
    """,
    equations = """
        tau * dalpha/dt + alpha = pos(post.r - 1.0) : postsynaptic
        eta * dw/dt = pre.r * post.r - alpha * post.r^2 * w : min=0.0
    """,
    name="Oja learning rule",
    description= """Oja learning rule ensuring regularization of the synaptic weights."""
)
  1. Choose simple parameter and variable names for the description of equations. If a parameter/variable name uses only one character, it will be treated as a mathematical variable in the equations (ex: v becomes \(v\)), otherwise the plain text representation will be used (ugly). If the name corresponds to a greek letter (alpha, tau, etc.), it will be represented by the corresponding greek letter (\(\alpha\), \(\tau\)). If the name is composed of two terms separated by an underscore (tau_exc), a subscript will be used (\(\tau_\text{exc}\)). If more than one underscore is used, the text representation is used instead (LaTeX does not allow multiple subscripts).

2.21.4. Example

Let’s take the homeostatic STDP ramp example provided in examples/homeostatic_stdp/Ramp.py and add names/descriptions to the objects:

from ANNarchy import *

# Izhikevich RS neuron
RSNeuron = Neuron(
    parameters = """
        a = 0.02 : population
        b = 0.2 : population
        c = -65. : population
        d = 8. : population
        tau_ampa = 5. : population
        tau_nmda = 150. : population
        vrev = 0.0 : population
    """ ,
    equations="""
        # Inputs
        I = g_ampa * (vrev - v) + g_nmda * nmda(v, -80.0, 60.0) * (vrev -v)
        # Midpoint scheme
        dv/dt = (0.04 * v + 5.0) * v + 140.0 - u + I : init=-65., midpoint
        du/dt = a * (b*v - u) : init=-13., midpoint
        # Izhikevich scheme
        # new_v = v + 0.5*(0.04 * v^2 + 5.0 * v + 140.0 - u + I) : init=-65.
        # v = new_v + 0.5*(0.04 * new_v^2 + 5.0 * new_v + 140.0 - u + I) : init=-65.
        # u += a * (b*v - u) : init=-13.
        # Conductances
        tau_ampa * dg_ampa/dt = -g_ampa : exponential
        tau_nmda * dg_nmda/dt = -g_nmda : exponential
    """ ,
    spike = """
        v >= 30.
    """,
    reset = """
        v = c
        u += d
    """,
    functions = """
        nmda(v, t, s) = ((v-t)/(s))^2 / (1.0 + ((v-t)/(s))^2)
    """,
    name = "Regular-spiking Izhikevich",
    description = "Regular-spiking Izhikevich neuron, with AMPA/NMDA exponentially decreasing synapses."
)

# Input population
inp = PoissonPopulation(100, rates=np.linspace(0.2, 20., 100), name="Poisson input")

# RS neuron without homeostatic mechanism
pop1 = Population(1, RSNeuron, name="RS neuron without homeostasis")
pop1.compute_firing_rate(5000.)

# RS neuron with homeostatic mechanism
pop2 = Population(1, RSNeuron, name="RS neuron with homeostasis")
pop2.compute_firing_rate(5000.)

# Nearest Neighbour STDP
nearest_neighbour_stdp = Synapse(
    parameters="""
        tau_plus = 20. : projection
        tau_minus = 60. : projection
        A_plus = 0.0002 : projection
        A_minus = 0.000066 : projection
        w_max = 0.03 : projection
    """,
    equations = """
        # Traces
        tau_plus  * dltp/dt = -ltp : exponential
        tau_minus * dltd/dt = -ltd : exponential
        # Nearest-neighbour
        w += if t_post >= t_pre: ltp else: - ltd : min=0.0, max=w_max
    """,
    pre_spike="""
        g_target += w
        ltp = A_plus
    """,
    post_spike="""
        ltd = A_minus
    """,
    name = "Nearest-neighbour STDP",
    description = "Nearest-neighbour STDP synaptic plasticity. Each synapse updates two traces (ltp and ltd) and updates continuously its weight."
)

# STDP with homeostatic regulation
homeo_stdp = Synapse(
    parameters="""
        # STDP
        tau_plus = 20. : projection
        tau_minus = 60. : projection
        A_plus = 0.0002 : projection
        A_minus = 0.000066 : projection
        w_min = 0.0 : projection
        w_max = 0.03 : projection

        # Homeostatic regulation
        alpha = 0.1 : projection
        beta = 1.0 : projection
        gamma = 50. : projection
        Rtarget = 35. : projection
        T = 5000. : projection
    """,
    equations = """
        # Traces
        tau_plus  * dltp/dt = -ltp : exponential
        tau_minus * dltd/dt = -ltd : exponential
        # Homeostatic values
        R = post.r : postsynaptic
        K = R/(T*(1.+fabs(1. - R/Rtarget) * gamma)) : postsynaptic
        # Nearest-neighbour
        stdp = if t_post >= t_pre: ltp else: - ltd
        w += (alpha * w * (1- R/Rtarget) + beta * stdp ) * K : min=w_min, max=w_max
    """,
    pre_spike="""
        g_target += w
        ltp = A_plus
    """,
    post_spike="""
        ltd = A_minus
    """ ,
    name = "Nearest-neighbour STDP with homeostasis",
    description = "Nearest-neighbour STDP synaptic plasticity with an additional homeostatic term. "
)

# Projection without homeostatic mechanism
proj1 = Projection(inp, pop1, ['ampa', 'nmda'], synapse=nearest_neighbour_stdp)
proj1.connect_all_to_all(Uniform(0.01, 0.03))

# Projection with homeostatic mechanism
proj2 = Projection(inp, pop2, ['ampa', 'nmda'], synapse=homeo_stdp)
proj2.connect_all_to_all(weights=Uniform(0.01, 0.03))


# Record
m1 = Monitor(pop1, 'r')
m2 = Monitor(pop2, 'r')

report('ramp.md',
        title="Biologically plausible models of homeostasis and STDP; Stability and learning in spiking neural networks",
        author="Carlson, Richert, Dutt and Krichmar",
        date="Neural Networks (IJCNN) 2013")

This generates the following Markdown file:

---
title: Biologically plausible models of homeostasis and STDP; Stability and learning in spiking neural networks
author: Carlson, Richert, Dutt and Krichmar
date: Neural Networks (IJCNN) 2013
---

# Structure of the network

* ANNarchy 4.6.2b using the default backend.
* Numerical step size: 1.0 ms.

## Populations

| **Population**                | **Size** | **Neuron type**            |
| ----------------------------- | -------- | -------------------------- |
| Poisson input                 | 100      | Poisson                    |
| RS neuron without homeostasis | 1        | Regular-spiking Izhikevich |
| RS neuron with homeostasis    | 1        | Regular-spiking Izhikevich |


## Projections

| **Source**    | **Destination**               | **Target**  | **Synapse type**                        | **Pattern**                                               |
| ------------- | ----------------------------- | ----------- | --------------------------------------- | --------------------------------------------------------- |
| Poisson input | RS neuron without homeostasis | ampa / nmda | Nearest-neighbour STDP                  | All-to-All, weights $\mathcal{U}$(0.01, 0.03), delays 0.0 |
| Poisson input | RS neuron with homeostasis    | ampa / nmda | Nearest-neighbour STDP with homeostasis | All-to-All, weights $\mathcal{U}$(0.01, 0.03), delays 0.0 |


## Monitors

| **Object**                    | **Variables** | **Period** |
| ----------------------------- | ------------- | ---------- |
| RS neuron without homeostasis | r             | 1.0        |
| RS neuron with homeostasis    | r             | 1.0        |


# Neuron models

## Regular-spiking Izhikevich

Regular-spiking Izhikevich neuron, with AMPA/NMDA exponentially decreasing synapses.

**Parameters:**

| **Name**             | **Default value** | **Locality**   | **Type** |
| -------------------- | ----------------- | -------------- | -------- |
| $a$                  | 0.02              | per population | double   |
| $b$                  | 0.2               | per population | double   |
| $c$                  | -65.0             | per population | double   |
| $d$                  | 8.0               | per population | double   |
| $\tau_{\text{ampa}}$ | 5.0               | per population | double   |
| $\tau_{\text{nmda}}$ | 150.0             | per population | double   |
| ${\text{vrev}}$      | 0.0               | per population | double   |

**Equations:**

* Variable $I$ : per neuron, initial value: 0.0

$$
{I}(t) = {g_{\text{ampa}}}(t) \cdot \left({\text{vrev}} - {v}(t)\right) + {g_{\text{nmda}}}(t) \cdot \left({\text{vrev}} - {v}(t)\right) \cdot \operatorname{nmda}{\left ({v}(t),-80.0,60.0 \right )}
$$

* Variable $v$ : per neuron, initial value: -65.0, midpoint numerical method

$$
\frac{d{v}(t)}{dt} = {I}(t) - {u}(t) + {v}(t) \cdot \left(0.04 \cdot {v}(t) + 5.0\right) + 140.0
$$

* Variable $u$ : per neuron, initial value: -13.0, midpoint numerical method

$$
\frac{d{u}(t)}{dt} = a \cdot \left(b \cdot {v}(t) - {u}(t)\right)
$$

* Variable $g_{\text{ampa}}$ : per neuron, initial value: 0.0, exponential numerical method

$$
\frac{d{g_{\text{ampa}}}(t)}{dt} \cdot \tau_{\text{ampa}} = - {g_{\text{ampa}}}(t)
$$

* Variable $g_{\text{nmda}}$ : per neuron, initial value: 0.0, exponential numerical method

$$
\frac{d{g_{\text{nmda}}}(t)}{dt} \cdot \tau_{\text{nmda}} = - {g_{\text{nmda}}}(t)
$$

**Spike emission:**

if ${v}(t) \geq 30.0$ :

* Emit a spike a time $t$.
* ${v}(t) = c$
* ${u}(t) \mathrel{+}= d$


**Functions**

$${\text{nmda}}(v, t, s) = \frac{\left(- t + v\right)^{2}}{s^{2} \cdot \left(1.0 + \frac{1}{s^{2}} \cdot \left(- t + v\right)^{2}\right)}$$


## Poisson

Spiking neuron with spikes emitted according to a Poisson distribution.

**Parameters:**

| **Name**         | **Default value** | **Locality** | **Type** |
| ---------------- | ----------------- | ------------ | -------- |
| ${\text{rates}}$ | 10.0              | per neuron   | double   |

**Equations:**

* Variable $p$ : per neuron, initial value: 0.0

$$
{p}(t) = \frac{1000.0}{\Delta t} \cdot \mathcal{U}{\left (0.0,1.0 \right )}
$$

**Spike emission:**

if ${p}(t) < {\text{rates}}$ :

* Emit a spike a time $t$.

# Synapse models

## Nearest-neighbour STDP

Nearest-neighbour STDP synaptic plasticity. Each synapse updates two traces (ltp and ltd) and updates continuously its weight.

**Parameters:**

| **Name**              | **Default value** | **Locality**   | **Type** |
| --------------------- | ----------------- | -------------- | -------- |
| $\tau_{\text{plus}}$  | 20.0              | per projection | double   |
| $\tau_{\text{minus}}$ | 60.0              | per projection | double   |
| $A_{\text{plus}}$     | 0.0002            | per projection | double   |
| $A_{\text{minus}}$    | 6.6e-05           | per projection | double   |
| $w_{\text{max}}$      | 0.03              | per projection | double   |

**Equations:**

* Variable ${\text{ltp}}$ : per synapse, initial value: 0.0, exponential numerical method

$$
\frac{d{{\text{ltp}}}(t)}{dt} \cdot \tau_{\text{plus}} = - {{\text{ltp}}}(t)
$$

* Variable ${\text{ltd}}$ : per synapse, initial value: 0.0, exponential numerical method

$$
\frac{d{{\text{ltd}}}(t)}{dt} \cdot \tau_{\text{minus}} = - {{\text{ltd}}}(t)
$$

* Variable $w$ : per synapse, initial value: 0.0, minimum: 0.0, maximum: w_max

$$
{w}(t) \mathrel{+}= \begin{cases}{{\text{ltp}}}(t)\qquad \text{if} \quad t_{\text{pos}} \geq t_{\text{pre}}\\ - {{\text{ltd}}}(t) \qquad \text{otherwise.} \end{cases}
$$

**Pre-synaptic event at $t_\text{pre} + d$:**
$$g_{\text{target}(t)} \mathrel{+}= {w}(t)$$
$${{\text{ltp}}}(t) = A_{\text{plus}}$$

**Post-synaptic event at $t_\text{post}$:**
$${{\text{ltd}}}(t) = A_{\text{minus}}$$

## Nearest-neighbour STDP with homeostasis

Nearest-neighbour STDP synaptic plasticity with an additional homeostatic term.

**Parameters:**

| **Name**              | **Default value** | **Locality**   | **Type** |
| --------------------- | ----------------- | -------------- | -------- |
| $\tau_{\text{plus}}$  | 20.0              | per projection | double   |
| $\tau_{\text{minus}}$ | 60.0              | per projection | double   |
| $A_{\text{plus}}$     | 0.0002            | per projection | double   |
| $A_{\text{minus}}$    | 6.6e-05           | per projection | double   |
| $w_{\text{min}}$      | 0.0               | per projection | double   |
| $w_{\text{max}}$      | 0.03              | per projection | double   |
| $\alpha$              | 0.1               | per projection | double   |
| $\beta$               | 1.0               | per projection | double   |
| $\gamma$              | 50.0              | per projection | double   |
| ${\text{Rtarget}}$    | 35.0              | per projection | double   |
| $T$                   | 5000.0            | per projection | double   |

**Equations:**

* Variable ${\text{ltp}}$ : per synapse, initial value: 0.0, exponential numerical method

$$
\frac{d{{\text{ltp}}}(t)}{dt} \cdot \tau_{\text{plus}} = - {{\text{ltp}}}(t)
$$

* Variable ${\text{ltd}}$ : per synapse, initial value: 0.0, exponential numerical method

$$
\frac{d{{\text{ltd}}}(t)}{dt} \cdot \tau_{\text{minus}} = - {{\text{ltd}}}(t)
$$

* Variable $R$ : per post-synaptic neuron, initial value: 0.0

$$
{R}(t) = {{r}^{\text{post}}}(t)
$$

* Variable $K$ : per post-synaptic neuron, initial value: 0.0

$$
{K}(t) = \frac{{R}(t)}{T \cdot \left(\gamma \cdot \left|{f}\right|{\left (- \frac{{R}(t)}{{\text{Rtarget}}} + 1.0 \right )} + 1.0\right)}
$$

* Variable ${\text{stdp}}$ : per synapse, initial value: 0.0

$$
{{\text{stdp}}}(t) = \begin{cases}{{\text{ltp}}}(t)\qquad \text{if} \quad t_{\text{pos}} \geq t_{\text{pre}}\\ - {{\text{ltd}}}(t) \qquad \text{otherwise.} \end{cases}
$$

* Variable $w$ : per synapse, initial value: 0.0, minimum: w_min, maximum: w_max

$$
{w}(t) \mathrel{+}= {K}(t) \cdot \left(\alpha \cdot {w}(t) \cdot \left(- \frac{{R}(t)}{{\text{Rtarget}}} + 1\right) + \beta \cdot {{\text{stdp}}}(t)\right)
$$

**Pre-synaptic event at $t_\text{pre} + d$:**
$$g_{\text{target}(t)} \mathrel{+}= {w}(t)$$
$${{\text{ltp}}}(t) = A_{\text{plus}}$$

**Post-synaptic event at $t_\text{post}$:**
$${{\text{ltd}}}(t) = A_{\text{minus}}$$

# Parameters

## Population parameters

| **Population**                | **Neuron type**            | **Name**             | **Value**     |
| ----------------------------- | -------------------------- | -------------------- | ------------- |
| Poisson input                 | Poisson                    | ${\text{rates}}$     | $[0.2, 20.0]$ |
| RS neuron without homeostasis | Regular-spiking Izhikevich | $a$                  | 0.02          |
|                               |                            | $b$                  | 0.2           |
|                               |                            | $c$                  | -65.0         |
|                               |                            | $d$                  | 8.0           |
|                               |                            | $\tau_{\text{ampa}}$ | 5.0           |
|                               |                            | $\tau_{\text{nmda}}$ | 150.0         |
|                               |                            | ${\text{vrev}}$      | 0.0           |
| RS neuron with homeostasis    | Regular-spiking Izhikevich | $a$                  | 0.02          |
|                               |                            | $b$                  | 0.2           |
|                               |                            | $c$                  | -65.0         |
|                               |                            | $d$                  | 8.0           |
|                               |                            | $\tau_{\text{ampa}}$ | 5.0           |
|                               |                            | $\tau_{\text{nmda}}$ | 150.0         |
|                               |                            | ${\text{vrev}}$      | 0.0           |


## Projection parameters

| **Projection**                                                                     | **Synapse type**                        | **Name**              | **Value** |
| ---------------------------------------------------------------------------------- | --------------------------------------- | --------------------- | --------- |
| Poisson input  $\rightarrow$ RS neuron without homeostasis with target ampa / nmda | Nearest-neighbour STDP                  | $\tau_{\text{plus}}$  | 20.0      |
|                                                                                    |                                         | $\tau_{\text{minus}}$ | 60.0      |
|                                                                                    |                                         | $A_{\text{plus}}$     | 0.0002    |
|                                                                                    |                                         | $A_{\text{minus}}$    | 6.6e-05   |
|                                                                                    |                                         | $w_{\text{max}}$      | 0.03      |
| Poisson input  $\rightarrow$ RS neuron with homeostasis with target ampa / nmda    | Nearest-neighbour STDP with homeostasis | $\tau_{\text{plus}}$  | 20.0      |
|                                                                                    |                                         | $\tau_{\text{minus}}$ | 60.0      |
|                                                                                    |                                         | $A_{\text{plus}}$     | 0.0002    |
|                                                                                    |                                         | $A_{\text{minus}}$    | 6.6e-05   |
|                                                                                    |                                         | $w_{\text{min}}$      | 0.0       |
|                                                                                    |                                         | $w_{\text{max}}$      | 0.03      |
|                                                                                    |                                         | $\alpha$              | 0.1       |
|                                                                                    |                                         | $\beta$               | 1.0       |
|                                                                                    |                                         | $\gamma$              | 50.0      |
|                                                                                    |                                         | ${\text{Rtarget}}$    | 35.0      |
|                                                                                    |                                         | $T$                   | 5000.0    |

Once transformed by pandoc, this generates for example the following html code:

Title:Biologically plausible models of homeostasis and STDP; Stability and learning in spiking neural networks
Author:Carlson, Richert, Dutt and Krichmar
Date:Neural Networks (IJCNN) 2013

1. Structure of the network

  • ANNarchy 4.6.3 using the default backend.
  • Numerical step size: 1.0 ms.

1.1 Populations

Population Size Neuron type
Poisson input 100 Poisson
RS neuron without homeostasis 1 Regular-spiking Izhikevich
RS neuron with homeostasis 1 Regular-spiking Izhikevich

1.2 Projections

Source Destination Target Synapse type Pattern
Poisson input RS neuron without homeostasis ampa / nmda Nearest-neighbour STDP All-to-All, weights \(\mathcal{U}\)(0.01, 0.03), delays 0.0
Poisso n input RS neuron with homeostasis ampa / nmda Nearest-neighbour STDP with homeostasis All-to-All, weights \(\mathcal{U}\)(0.01, 0.03), delays 0.0

1.3 Monitors

Object Variables Period
RS neuron without homeostasis r 1.0
RS neuron with homeostasis r 1.0

2 Neuron models

2.1 Regular-spiking Izhikevich

Regular-spiking Izhikevich neuron, with AMPA/NMDA exponentially decreasing synapses.

Parameters:

Name Default value Locality Type
\(a\) 0.02 per population double
\(b\) 0.2 per population double
\(c\) -65.0 per population double
\(d\) 8.0 per population double
\(\tau_{\text{ampa}}\) 5.0 per population double
\(\tau_{\text{nmda}}\) 150.0 per population double
\({\text{vrev}}\) 0.0 per population double

Equations:

  • Variable \(I\) : per neuron, initial value: 0.0
\[{I}(t) = {g_{\text{ampa}}}(t) \cdot \left({\text{vrev}} - {v}(t)\right) + {g_{\text{nmda}}}(t) \cdot \left({\text{vrev}} - {v}(t)\right) \cdot \operatorname{nmda}{\left ({v}(t),-80.0,60.0 \right )}\]
  • Variable \(v\) : per neuron, initial value: -65.0, midpoint numerical method
\[\frac{d{v}(t)}{dt} = {I}(t) - {u}(t) + {v}(t) \cdot \left(0.04 \cdot {v}(t) + 5.0\right) + 140.0\]
  • Variable \(u\) : per neuron, initial value: -13.0, midpoint numerical method
\[\frac{d{u}(t)}{dt} = a \cdot \left(b \cdot {v}(t) - {u}(t)\right)\]
  • Variable \(g_{\text{ampa}}\) : per neuron, initial value: 0.0, exponential numerical method
\[\frac{d{g_{\text{ampa}}}(t)}{dt} \cdot \tau_{\text{ampa}} = - {g_{\text{ampa}}}(t)\]
  • Variable \(g_{\text{nmda}}\) : per neuron, initial value: 0.0, exponential numerical method
\[\frac{d{g_{\text{nmda}}}(t)}{dt} \cdot \tau_{\text{nmda}} = - {g_{\text{nmda}}}(t)\]

Spike emission:

if \({v}(t) \geq 30.0\) :

  • Emit a spike a time \(t\).
  • \({v}(t) = c\)
  • \({u}(t) \mathrel{+}= d\)

Functions

\[{\text{nmda}}(v, t, s) = \frac{\left(- t + v\right)^{2}}{s^{2} \cdot \left(1.0 + \frac{1}{s^{2}} \cdot \left(- t + v\right)^{2}\right)}\]

2.2 Poisson

Spiking neuron with spikes emitted according to a Poisson distribution.

Parameters:

Name Default value Locality Type
\({\text{rates}}\) 10.0 per neuron double

Equations:

  • Variable \(p\) : per neuron, initial value: 0.0
\[{p}(t) = \frac{1000.0}{\Delta t} \cdot \mathcal{U}{\left (0.0,1.0 \right )}\]

Spike emission:

if \({p}(t) < {\text{rates}}\) :

  • Emit a spike a time \(t\).

3 Synapse models

3.1 Nearest-neighbour STDP

Nearest-neighbour STDP synaptic plasticity. Each synapse updates two traces (ltp and ltd) and updates continuously its weight.

Parameters:

Name Default value Locality Type
\(\tau_{\text{plus}}\) 20.0 per projection double
\(\tau_{\text{minus}}\) 60.0 per projection double
\(A_{\text{plus}}\) 0.0002 per projection double
\(A_{\text{minus}}\) 6.6e-05 per projection double
\(w_{\text{max}}\) 0.03 per projection double

Equations:

  • Variable \({\text{ltp}}\) : per synapse, initial value: 0.0, exponential numerical method
\[\frac{d{{\text{ltp}}}(t)}{dt} \cdot \tau_{\text{plus}} = - {{\text{ltp}}}(t)\]
  • Variable \({\text{ltd}}\) : per synapse, initial value: 0.0, exponential numerical method
\[\frac{d{{\text{ltd}}}(t)}{dt} \cdot \tau_{\text{minus}} = - {{\text{ltd}}}(t)\]
  • Variable \(w\) : per synapse, initial value: 0.0, minimum: 0.0, maximum: w_max
\[\begin{split}{w}(t) \mathrel{+}= \begin{cases}{{\text{ltp}}}(t)\qquad \text{if} \quad t_{\text{pos}} \geq t_{\text{pre}}\\ - {{\text{ltd}}}(t) \qquad \text{otherwise.} \end{cases}\end{split}\]

Pre-synaptic event at :math:`t_text{pre} + d`:

\[g_{\text{target}(t)} \mathrel{+}= {w}(t)\]
\[{{\text{ltp}}}(t) = A_{\text{plus}}\]

Post-synaptic event at :math:`t_text{post}`:

\[{{\text{ltd}}}(t) = A_{\text{minus}}\]

3.2 Nearest-neighbour STDP with homeostasis

Nearest-neighbour STDP synaptic plasticity with an additional homeostatic term.

Parameters:

Name Default value Locality Type
\(\tau_{\text{plus}}\) 20.0 per projection double
\(\tau_{\text{minus}}\) 60.0 per projection double
\(A_{\text{plus}}\) 0.0002 per projection double
\(A_{\text{minus}}\) 6.6e-05 per projection double
\(w_{\text{min}}\) 0.0 per projection double
\(w_{\text{max}}\) 0.03 per projection double
\(\alpha\) 0.1 per projection double
\(\beta\) 1.0 per projection double
\(\gamma\) 50.0 per projection double
\({\text{Rtarget}}\) 35.0 per projection double
\(T\) 5000.0 per projection double

Equations:

  • Variable \({\text{ltp}}\) : per synapse, initial value: 0.0, exponential numerical method
\[\frac{d{{\text{ltp}}}(t)}{dt} \cdot \tau_{\text{plus}} = - {{\text{ltp}}}(t)\]
  • Variable \({\text{ltd}}\) : per synapse, initial value: 0.0, exponential numerical method
\[\frac{d{{\text{ltd}}}(t)}{dt} \cdot \tau_{\text{minus}} = - {{\text{ltd}}}(t)\]
  • Variable \(R\) : per post-synaptic neuron, initial value: 0.0
\[{R}(t) = {{r}^{\text{post}}}(t)\]
  • Variable \(K\) : per post-synaptic neuron, initial value: 0.0
\[{K}(t) = \frac{{R}(t)}{T \cdot \left(\gamma \cdot \left|{f}\right|{\left (- \frac{{R}(t)}{{\text{Rtarget}}} + 1.0 \right )} + 1.0\right)}\]
  • Variable \({\text{stdp}}\) : per synapse, initial value: 0.0
\[\begin{split}{{\text{stdp}}}(t) = \begin{cases}{{\text{ltp}}}(t)\qquad \text{if} \quad t_{\text{pos}} \geq t_{\text{pre}}\\ - {{\text{ltd}}}(t) \qquad \text{otherwise.} \end{cases}\end{split}\]
  • Variable \(w\) : per synapse, initial value: 0.0, minimum: w_min, maximum: w_max
\[{w}(t) \mathrel{+}= {K}(t) \cdot \left(\alpha \cdot {w}(t) \cdot \left(- \frac{{R}(t)}{{\text{Rtarget}}} + 1\right) + \beta \cdot {{\text{stdp}}}(t)\right)\]

Pre-synaptic event at :math:`t_text{pre} + d`:

\[g_{\text{target}(t)} \mathrel{+}= {w}(t)\]
\[{{\text{ltp}}}(t) = A_{\text{plus}}\]

Post-synaptic event at :math:`t_text{post}`:

\[{{\text{ltd}}}(t) = A_{\text{minus}}\]

4 Parameters

4.1 Population parameters

Population Neuron type Name Value
Poisson input Poisson \({\text{rates}}\) \([0.2, 20.0]\)
RS neuron without homeostasis Regular-spiking Izhikevich \(a\) 0.02
    \(b\) 0.2
    \(c\) -65.0
    \(d\) 8.0
    \(\tau_{\text{ampa}}\) 5.0
    \(\tau_{\text{nmda}}\) 150.0
    \({\text{vrev}}\) 0.0
RS neuron with homeostasis Regular-spiking Izhikevich \(a\) 0.02
    \(b\) 0.2
    \(c\) -65.0
    \(d\) 8.0
    \(\tau_{\text{ampa}}\) 5.0
    \(\tau_{\text{nmda}}\) 150.0
    \({\text{vrev}}\) 0.0

4.2 Projection parameters

Projection Synapse type Name Value
Poisson input \(\rightarrow\) RS neuron without homeostasis with target ampa / nmda Nearest-neighbour STDP \(\tau_{\text{plus}}\) 20.0
    \(\tau_{\text{minus}}\) 60.0
    \(A_{\text{plus}}\) 0.0002
    \(A_{ \text{minus}}\) 6.6e-5
    \(w_{ \text{max}}\) 0.03
Poisson input \(\rightarrow\) RS neuron with homeostasis with target ampa / nmda Nearest-neighbour STDP with homeostasis \(\tau_{\text{plus}}\) 20.0
    \(\tau_{\text{minus}}\) 60.0
    \(A_{\text{plus}}\) 0.0002
    \(A_{\text{minus}}\) 6.6e-5
    \(w_{\text{min}}\) 0.0
    \(w_{\text{max}}\) 0.03
    \(\alpha\) 0.1
    \(\beta\) 1.0
    \(\gamma\) 50.0
    \({\text{Rtarget}}\) 35.0
    \(T\) 5000