ANNarchy
Artificial Neural Networks architect
Computational neuroscience
Computational neuroscience is about explaining brain functioning at various levels (neural activity patterns, behavior, etc.) using biologically realistic neuro-computational models.
Different types of neural and synaptic mathematical models are used in the field, abstracting biological complexity at different levels.
There is no “right” level of biological plausibility for a model - you can always add more details -, but you have to find a paradigm that allows you to:
- Explain the current experimental data.
- Make predictions that can be useful to experimentalists.
Rate-coded and spiking neurons
- Rate-coded neurons only represent the instantaneous firing rate of a neuron:
\tau \, \frac{d v(t)}{dt} + v(t) = \sum_{i=1}^d w_{i, j} \, r_i(t) + b
r(t) = f(v(t))
- Spiking neurons emit binary spikes when their membrane potential exceeds a threshold (leaky integrate-and-fire, LIF):
C \, \frac{d v(t)}{dt} = - g_L \, (v(t) - V_L) + I(t)
\text{if} \; v(t) > V_T \; \text{emit a spike and reset.}
Many different spiking neuron models are possible
- Izhikevich quadratic IF (Izhikevich, 2003).
\begin{cases} \displaystyle\frac{dv}{dt} = 0.04 \, v^2 + 5 \, v + 140 - u + I \\ \\ \displaystyle\frac{du}{dt} = a \, (b \, v - u) \\ \end{cases}
- Adaptive exponential IF (AdEx, Brette and Gerstner, 2005).
\begin{cases} \begin{aligned} C \, \frac{dv}{dt} = -g_L \ (v - E_L) + & g_L \, \Delta_T \, \exp(\frac{v - v_T}{\Delta_T}) \\ & + I - w \end{aligned}\\ \\ \tau_w \, \displaystyle\frac{dw}{dt} = a \, (v - E_L) - w\\ \end{cases}
Realistic neuron models can reproduce a variety of dynamics
Biological neurons do not all respond the same to an input current: Some fire regularly, some slow down with time., some emit bursts of spikes…
Modern spiking neuron models allow to recreate these dynamics by changing just a few parameters.
Populations of neurons
- Recurrent neural networks (e.g. randomly connected populations of neurons) can exhibit very rich dynamics even in the absence of inputs:
Oscillations at the population level.
Excitatory/inhibitory balance.
Spatio-temporal separation of inputs (reservoir computing).
Synaptic plasticity: Hebbian learning
- Hebbian learning postulates that synapses strengthen based on the correlation between the activity of the pre- and post-synaptic neurons:
When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.
Donald Hebb, 1949
- Weights increase proportionally to the the product of the pre- and post-synaptic firing rates:
\frac{dw}{dt} = \eta \, r^\text{pre} \, r^\text{post}
Synaptic plasticity: Hebbian-based learning
- The BCM (Bienenstock et al., 1982; Intrator and Cooper, 1992) plasticity rule allows LTP and LTD depending on the post-synaptic activity:
\frac{dw}{dt} = \eta \, r^\text{pre} \, r^\text{post} \, (r^\text{post} - \mathbb{E}[(r^\text{post})^2])
- Covariance learning rule (Dayan and Abbott, 2001):
\frac{dw}{dt} = \eta \, r^\text{pre} \, (r^\text{post} - \mathbb{E}[r^\text{post}])
- Oja learning rule (Oja, 1982):
\frac{dw}{dt}= \eta \, r^\text{pre} \, r^\text{post} - \alpha \, (r^\text{post})^2 \, w
or virtually anything depending only on the pre- and post-synaptic firing rates, e.g. Vitay and Hamker (2010):
\begin{aligned} \frac{dw}{dt} & = \eta \, ( \text{DA}(t) - \overline{\text{DA}}) \, (r^\text{post} - \mathbb{E}[r^\text{post}] )^+ \, (r^\text{pre} - \mathbb{E}[r^\text{pre}])- \alpha(t) \, ((r^\text{post} - \mathbb{E}[r^\text{post}] )^+ )^2 \, w \end{aligned}
STDP: Spike-timing dependent plasticity
Synaptic efficiencies actually evolve depending on the the causation between the neuron’s firing patterns:
If the pre-synaptic neuron fires before the post-synaptic one, the weight is increased (long-term potentiation). Pre causes Post to fire.
If it fires after, the weight is decreased (long-term depression). Pre does not cause Post to fire.
STDP: Spike-timing dependent plasticity
- The STDP (spike-timing dependent plasticity, Bi and Poo, 2001) plasticity rule describes how the weight of a synapse evolves when the pre-synaptic neuron fires at t_\text{pre} and the post-synaptic one fires at t_\text{post}.
\frac{dw}{dt} = \begin{cases} A^+ \, \exp - \frac{t_\text{pre} - t_\text{post}}{\tau^+} \; \text{if} \; t_\text{post} > t_\text{pre}\\ \\ A^- \, \exp - \frac{t_\text{pre} - t_\text{post}}{\tau^-} \; \text{if} \; t_\text{pre} > t_\text{post}\\ \end{cases}
STDP can be implemented online using traces.
More complex variants of STDP (triplet STDP) exist, but this is the main model of synaptic plasticity in spiking networks.
Neuro-computational modeling
Populations of neurons can be combined in functional neuro-computational models learning to solve various tasks.
Need to implement one (or more) equations per neuron and synapse. There can thousands of neurons and millions of synapses, see Teichmann et al. (2021).
ANNarchy (Artificial Neural Networks architect)
Vitay et al. (2015)
ANNarchy: a code generation approach to neural simulations on parallel hardware.
Frontiers in Neuroinformatics 9. doi:10.3389/fninf.2015.00019
Features
Simulation of both rate-coded and spiking neural networks.
Only local biologically realistic mechanisms are possible (no backpropagation).
Equation-oriented description of neural/synaptic dynamics (à la Brian).
Code generation in C++, parallelized using OpenMP on CPU and CUDA on GPU (MPI is coming).
Synaptic, intrinsic and structural plasticity mechanisms.
Echo-State Network
- ESN rate-coded neurons follow first-order ODEs:
\tau \frac{dx(t)}{dt} + x(t) = \sum w^\text{in} \, r^\text{in}(t) + g \, \sum w^\text{rec} \, r(t) + \xi(t)
r(t) = \tanh(x(t))
- Neural dynamics are described by the equation-oriented interface:
import ANNarchy as ann
ESN_Neuron = ann.Neuron(
parameters = """
tau = 30.0 : population # Time constant
g = 1.0 : population # Scaling
noise = 0.01 : population # Noise level
""",
equations="""
tau * dx/dt + x = sum(in) + g * sum(exc) + noise * Uniform(-1, 1) : init=0.0
r = tanh(x)
"""
)Notebook: Echo-State Network
\tau \frac{dx(t)}{dt} + x(t) = \sum_\text{input} W^\text{IN} \, r^\text{IN}(t) + g \, \sum_\text{rec} W^\text{REC} \, r(t) + \xi(t)
r(t) = \tanh(x(t))
Notebook: IBCM learning rule
Variant of the BCM (Bienenstock, Cooper, Munro, 1982) learning rule.
The LTP and LTD depend on post-synaptic activity: homeostasis.
\Delta w = \eta \, r^\text{pre} \, r^\text{post} \, (r^\text{post} - \mathbb{E}[(r^\text{post})^2])
Notebook: Reward-modulated RC network of Miconi (2017)
e_{i, j}(t) = e_{i, j}(t-1) + (r_i (t) \, x_j(t))^3
\Delta w_{i, j} = - \eta \, e_{i, j}(T) \, (R -R_\text{mean})
Spiking neurons
Spiking neurons must also define two additional fields:
spike: condition for emitting a spike.reset: what happens after a spike is emitted (at the start of the refractory period).
A refractory period in ms can also be specified.
- Example of the Leaky Integrate-and-Fire:
C \, \frac{d v(t)}{dt} = - g_L \, (v(t) - V_L) + I(t)
\text{if} \; v(t) > V_T \; \text{emit a spike and reset.}
LIF = ann.Neuron(
parameters = """
C = 200.
g_L = 10.
E_L = -70.
v_T = 0.
v_r = -58.
I = 0.25
""",
equations = """
C * dv/dt = g_L * (E_L - v) + I : init=E_L
""",
spike = "v >= v_T",
reset = "v = v_r",
refractory = 2.0
)Notebook: AdEx neuron - Adaptive exponential Integrate-and-fire
\tau \cdot \frac{dv (t)}{dt} = E_l - v(t) + g_\text{exc} (t) \, (E_\text{exc} - v(t)) + g_\text{inh} (t) \, (E_\text{inh} - v(t)) + I(t)
Conductances / currents
- A pre-synaptic spike arriving to a spiking neuron increases the conductance/current
g_target(e.g.g_excorg_inh, depending on the projection).
LIF = Neuron(
parameters = "...",
equations = """
C*dv/dt = g_L*(E_L - v) + g_exc : init=E_L
""",
spike = "v >= v_T",
reset = "v = v_r",
refractory = 2.0
)- Each spike increments instantaneously
g_targetfrom the synaptic efficiencywof the corresponding synapse.
g_target += w
Conductances / currents
For exponentially-decreasing or alpha-shaped synapses, ODEs have to be introduced for the conductance/current.
The exponential numerical method should be preferred, as integration is exact.
LIF = ann.Neuron(
parameters = "...",
equations = """
C*dv/dt = g_L*(E_L - v) + g_exc : init=E_L
tau_exc * dg_exc/dt = - g_exc : exponential
""",
spike = "v >= v_T",
reset = "v = v_r",
refractory = 2.0
)
Notebook: Synaptic transmission
# Membrane potential
tau * dv/dt = (E_L - v) + g_a + g_b + alpha_c
# Exponentially decreasing
tau_b * dg_b/dt = -g_b
# Alpha-shaped
tau_c * dg_c/dt = -g_c
tau_c * dh_c/dt = g_c - h_c
Notebook: COBA - Conductance-based E/I network
\tau \, \frac{dv (t)}{dt} = E_l - v(t) + g_\text{exc} (t) \, (E_\text{exc} - v(t)) + g_\text{inh} (t) \, (E_\text{inh} - v(t)) + I(t)
Notebook: STP
Notebook: STDP
\begin{cases} \tau^+ \, \dfrac{d x(t)}{dt} = -x(t) \\ \\ \tau^- \, \dfrac{d y(t)}{dt} = -y(t) \\ \end{cases}
