# Quickstart¶

Contents

Two main functions are provided in BrainPy: neurodynamics simulation and neurodynamics analysis. In this part, I will focus on neuronal dynamics simulation, and tell you how to code a dynamical network in BrainPy bu using the example of (Wang & Buzsáki, 1996).

• Wang, Xiao-Jing, and György Buzsáki. “Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model.” Journal of neuroscience 16.20 (1996): 6402-6413.

(Wang & Buzsáki, 1996) demonstrates how a group of neuron with mutual inhibition produce the gamma oscillation (20–80 Hz) observed in the neocortex and hippocampus. In this network model, the neurons are modeled as a variant of Hodgkin–Huxley (HH) neuron model, and the inhibition connections between neurons are modeled as the GABAA synapses.

Here, we will first build a HH neuron model. Then, construct a GABAA synapse model. Finally, combining the HH model and GABAA moldel together, we will build a network model. We expect at the suitable parameter regions, the network will produce gamma oscillation.

First of all, import your favorite brainpy and numpy package.

[1]:

import brainpy as bp
import numpy as np


In BrainPy, all the system-level settings are implmented in bp.profile. By using bp.profile, you can set the backend or the device of the models going to run on, the method and the precision of the numerical integrator, etc. Before diving into this tutorial, let’s set the necessary profiles:

[2]:

bp.profile.set(jit=True,
device='cpu',
dt=0.04,
numerical_method='exponential')


This setting means we will JIT compile our model on cpu device, the default numerical method is set to exponential euler method (exponential), and the numerical step is set to 0.04.

## How to build a neuron model?¶

In BrainPy, the solving of differential equations is based on numerical methods, such as Euler method, Runge–Kutta methods. Therefore, the definition of a neuron/synapse model is the definition of the step functions which explicitly point out how variable states at the current time $$x(t)$$ is converted to the next time $$x(t+1)$$.

Let’s take the neuron model as an example.

To build a neuron model in BrainPy is to create an instance of NeuType. The instantiation of NeuType requires three items:

• ST: The neuron model state.

• steps: The step function to update at each cycle of run.

• name : The name of the neuron model (will be useful in error reporting and debugging).

Here, we are going to create a HH neuron model. The parameters of HH model are defined in the follows:

[3]:

V_th = 0.  # the spike threshold
C = 1.0  # the membrane capacitance
gLeak = 0.1  # the conductance of leaky channel
ELeak = -65  # the reversal potential of the leaky channel
gNa = 35.  # the conductance of sodium channel
ENa = 55.  # the reversal potential of sodium
gK = 9.  # the conductance of potassium channel
EK = -90.  # the reversal potential of potassium
phi = 5.0  # the temperature depdendent scaling


In this variant of HH model, three dynamical variables ($$V$$, $$h$$ and $$n$$) exist.

Coding the differential equations

${dx \over dt} = f(x, t)$

you only need write down the right-hand part of the differential equations $$f(x, t)$$. By adding a powerfull decorator porovided by BrainPy, @bp.integrate, the framework will automatically numerically integrate the defined equations. Generally, an ordinary differential equation in BrainPy can be coded as:

[4]:

@bp.integrate
def func(x, t, other_arguments):
# ... some computation ...
dxdt = ...
return dxdt


@bp.integrate receives method keyword to specify the numerical method you want to choose. For example, adding @bp.integrate(method='rk4') means you integrate the decorated function by using Fouth-order Runge–Kutta method (The full list of supportted numerical integrators please see the document of Numerical integrators). Otherwise, the differential function will be integrated by the system default method.

Specifically, for h channel,

\begin{split}\begin{align} {\frac {dh}{dt}} &=\alpha _{h}(V)(1-h)-\beta _{h}(V)h \\ \alpha_h(V) &= 0.07 \cdot \exp\big(-{V+58 \over 20}\big) \\ \beta_h(V) &= {1 \over 1 + \exp\big(-{V+28\over 10}\big)} \end{align}\end{split}

you can code it like this with BrainPy:

[5]:

@bp.integrate
def int_h(h, t, V):
alpha = 0.07 * np.exp(-(V + 58) / 20)
beta = 1 / (np.exp(-0.1 * (V + 28)) + 1)
dhdt = alpha * (1 - h) - beta * h
return phi * dhdt


The differential equation of n channel

\begin{split}\begin{align} {\frac {dn}{dt}} &=\alpha _{n}(V)(1-n)-\beta _{n}(V)n \\ \alpha_n(V) &= {0.01 \cdot (V+34) \over 1-\exp\big(-{V+34 \over10}\big)} \\ \beta_n(V) &= 0.125 \cdot \exp\big(-{V+44 \over 80}\big) \end{align}\end{split}

can be coded as:

[6]:

@bp.integrate
def int_n(n, t, V):
alpha = -0.01 * (V + 34) / (np.exp(-0.1 * (V + 34)) - 1)
beta = 0.125 * np.exp(-(V + 44) / 80)
dndt = alpha * (1 - n) - beta * n
return phi * dndt


Finally, the differential equation of membrane potential V is expressed as:

\begin{align} C_{m}{\frac {d V}{dt}}&=-{\bar {g}}_{\text{K}}n^{4}(V-V_{K}) - {\bar {g}}_{\text{Na}}m^{3}h(V-V_{Na}) -{\bar {g}}_{l}(V-V_{l}) + I_{syn} \end{align}

where $$m$$ is modeled as an instaneous channel (fast enough and substituted by its steady-state function)

\begin{split}\begin{align} \alpha_m(V) &= {0.1 (V+ 40) \over 1-\exp\big(-{ V+40 \over 10}\big)} \\ \beta_m(V) &= 4.0 \cdot \exp\big(-{V+65 \over 18}\big) \\ m &= {\alpha_m(V) \over \alpha_m(V) + \beta_m(V)} \end{align}\end{split}

Therefore, the differential equations of $$V$$ is coded as:

[7]:

@bp.integrate
def int_V(V, t, h, n, Isyn):
m_alpha = -0.1 * (V + 35) / (np.exp(-0.1 * (V + 35)) - 1)
m_beta = 4 * np.exp(-(V + 60) / 18)
m = m_alpha / (m_alpha + m_beta)
INa = gNa * m ** 3 * h * (V - ENa)
IK = gK * n ** 4 * (V - EK)
IL = gLeak * (V - ELeak)
dvdt = (- INa - IK - IL + Isyn) / C
return dvdt


Neuron state

In order to support the convenient state management, BrainPy provides NeuState to help you manage your model state.

In HH neuron, there are $$V$$, $$h$$ and $$n$$ dynamical variables. Moreover, we can add a input item in HH neuron state to receive the varying input. Further, the neuron spike is also we take care of. So, the neuron state of HH model can be specified as

[8]:

HH_ST = bp.types.NeuState(
V=-55.,  # membrane potential, default initial value is -55.
h=0.,  # h channel, default initial value is 0.
n=0.,  # n channel, default initial value is 0.
spike=0.,  # neuron spike state, default initial value is 0.,
# if neuron emits a spike, it will be 1.
input=0.  # neuron synaptic input, default initial value is 0.
)


The instantiation of bp.types.NeuState can receive strings (*args), or key=value pairs (**kwargs). It can also be instantiated by a dict (which means the fields and their default initial values), or a list/tuple of fields (in this case the default initial value will be set to 0.).

Step functions

For each model, the most important thing is to define the step functions. After the definition of differential equations, the step function of the HH model can be defined as:

[9]:

def update(ST, _t):
h = int_h(ST['h'], _t, ST['V'])
n = int_n(ST['n'], _t, ST['V'])
V = int_V(ST['V'], _t, ST['h'], ST['n'], ST['input'])
spike = np.logical_and(ST['V'] < V_th, V >= V_th)
ST['spike'] = spike
ST['V'] = V
ST['h'] = h
ST['n'] = n
ST['input'] = 0.


To define a step function, you can pass any data you need into the function as the functional arguments. The order of the arguments in each step function can be arbitrary.

In HH model step function, as you can see, two arguments are required:

• ST: the neuron state.

• _t: the current time, which is a system keyword, and denotes the current time point. In BrainPy, there are three system keywords: _t, _i (the current running step number), and _dt (the numerical integration precision).

NeuType

Finally, putting the above together, we get our HH neuron model as:

[10]:

HH = bp.NeuType(ST=HH_ST, name='HH_neuron', steps=update)


More advanced usage of NeuType definition please see Build Neurons.

## How to build a synapse model?¶

Like the NeuType definition, let’s announce the parameters all we need in the following:

[11]:

g_max = 0.1  # the maximal synaptic conductance
E = -75.  # the reversal potential
alpha = 12.  # the channel opening rate
beta = 0.1  # the channel closing rate


The GABAA synapse defined in (Wang & Buzsáki, 1996) is mathematically expressed as

\begin{split}\begin{align} \frac{ds}{dt} &= \alpha F(V_{pre})(1-s)-\beta s \quad (1) \\ F(V_{pre}) &= {1 \over 1+\exp\left(-{V_{pre}- V_{th} \over 2}\right)} \quad (2) \end{align}\end{split}

The synaptic current output onto the post-synaptic neuron is expressed as

$I_{ syn }=g_{max} s\left(V-E\right) \quad (3)$

Obviously, GABAA synapse model has one dynamical variable s. Thus, we can create the synapse state by using

[12]:

ST = bp.types.SynState('s')  # the initial (default) value is 0.


Based on the equation (1) and (2), the state updating of GABAA synapse is coded as:

[13]:

@bp.integrate
def int_s(s, t, TT):
return alpha * TT * (1 - s) - beta * s

def update(ST, _t, pre):
T = 1 / (1 + np.exp(-(pre['V'] - V_th) / 2))
s = int_s(ST['s'], _t, T)
ST['s'] = s


Moreover, based on the equation (3), the delayed synaptic value output onto the post-synaptic neurons of GABAA synpase can be coded as:

[14]:

@bp.delayed
def output(ST, post):
post['input'] -= g_max * ST['s'] * (post['V'] - E)


The decorator @bp.delayed can be added on the function which need the delayed ST. BrainPy will automatically recognize the delayed fileds. For example, in this output() function, the field s will be automatically delayed. When calling output(), the ST['s'] will be the delayed one.

Moreover, as you can see, in addition to ST and _t, the definition of the GABAA synapse requires pre and post (declared as function arguments in output() and update()). If you use this model defined by yourself, you clearly know what data you need to run the model. However, when somebody use your defined model, they will be confused by what pre and post mean. So, here we can make type declarations (optional):

[15]:

requires = dict(
pre=bp.types.NeuState(['V'], help='pre-synaptic neuron state'),
post=bp.types.NeuState(['V', 'input'], help='post-synaptic neuron state'),
)


which means this model need a pre-synaptic “NeuState” data pre (in which the field V is needed to compute the synapatic state) and a post-synaptic “NeuState” post (in which the fields V and input are needed).

Finally, let’s put the above difinitions together, and we get our wantted synapse model:

[16]:

GABAa = bp.SynType(ST=ST,
name='GABAa',
steps=(update, output),
requires=requires,
mode='scalar')


## How to construct a network?¶

It is worthy to note that the above defined HH NeuType and GABAa SynType are abstract models. They can not be used for concrete computation. Instead, we should define the NeuGroup and SynConn.

Here, by using bp.NeuGroup, let’s define a neuron group which contains 100 neurons. At the same time, we monitor the history trajectory of membrane potential V and spikes spike.

[17]:

num = 100
neu = bp.NeuGroup(HH, geometry=num, monitors=['spike', 'V'])


Similarly, the concrete synaptic connection can be constructed by using bp.SynConn. It receives an instance of SynType (argument model), the pre-synaptic neuron group (argument pre_group), the post-synaptic neuron group (argument post_group), the connection methods between the two groups (argument conn), and the delay length (argument delay).

[18]:

syn = bp.SynConn(model=GABAa,
pre_group=neu,
post_group=neu,
conn=bp.connect.All2All(include_self=False),
delay=0.5,
monitors=['s'])


The initial state value of a neuron group or an ensemble of synaptical connections can be updated by set neu_group.ST[key] = value, or syn_conn.ST[key] = value. In this example, we can update the initial value of neu as:

[19]:

v_init = -70. + np.random.random(num) * 20
h_alpha = 0.07 * np.exp(-(v_init + 58) / 20)
h_beta = 1 / (np.exp(-0.1 * (v_init + 28)) + 1)
h_init = h_alpha / (h_alpha + h_beta)
n_alpha = -0.01 * (v_init + 34) / (np.exp(-0.1 * (v_init + 34)) - 1)
n_beta = 0.125 * np.exp(-(v_init + 44) / 80)
n_init = n_alpha / (n_alpha + n_beta)

neu.ST['V'] = v_init
neu.ST['h'] = h_init
neu.ST['n'] = n_init


Moreover, the parameters of the created neuron groups or synaptical connections can be updated by using neu_group.pars[key] = value, or syn_conn.pars[key] = value. In this example, we can update the parameter of syn as:

[20]:

syn.pars['g_max'] = 0.1 / num


Finally, by adding the created neuron groups and synapse connection into the bp.Network, we get an instance of the network. Each bp.Network has a powerful function .run(). You can specify the total duration to run (argument duration), the inputs to various components (argument inputs), and the option for progress reporting (arguments report and report_percent).

[21]:

net = bp.Network(neu, syn)
net.run(duration=500., inputs=[neu, 'ST.input', 1.2], report=False)


Let’s visualize the network running results.

[22]:

import matplotlib.pyplot as plt

ts = net.ts
fig, gs = bp.visualize.get_figure(2, 1, 3, 12)

plt.plot(ts, neu.mon.V[:, 0])
plt.ylabel('Membrane potential (N0)')
plt.xlim(net.t_start - 0.1, net.t_end + 0.1)