from __future__ import division
import logging
import warnings
import numpy as np
from nengo.exceptions import SimulationError, ValidationError
from nengo.params import Parameter, NumberParam, FrozenObject
from nengo.utils.compat import range
from nengo.utils.neurons import settled_firingrate
logger = logging.getLogger(__name__)
[docs]class NeuronType(FrozenObject):
"""Base class for Nengo neuron models.
Attributes
----------
probeable : tuple
Signals that can be probed in the neuron population.
"""
probeable = ()
def __repr__(self):
return "%s(%s)" % (type(self).__name__, ", ".join(self._argreprs))
@property
def _argreprs(self):
return []
[docs] def current(self, x, gain, bias):
"""Compute current injected in each neuron given input, gain and bias.
Parameters
----------
x : (n_neurons,) array_like
Vector-space input.
gain : (n_neurons,) array_like
Gains associated with each neuron.
bias : (n_neurons,) array_like
Bias current associated with each neuron.
"""
x = np.array(x, dtype=float, copy=False, ndmin=1)
gain = np.array(gain, dtype=float, copy=False, ndmin=1)
bias = np.array(bias, dtype=float, copy=False, ndmin=1)
return gain * x + bias
[docs] def gain_bias(self, max_rates, intercepts):
"""Compute the gain and bias needed to satisfy max_rates, intercepts.
This takes the neurons, approximates their response function, and then
uses that approximation to find the gain and bias value that will give
the requested intercepts and max_rates.
Note that this default implementation is very slow! Whenever possible,
subclasses should override this with a neuron-specific implementation.
Parameters
----------
max_rates : (n_neurons,) array_like
Maximum firing rates of neurons.
intercepts : (n_neurons,) array_like
X-intercepts of neurons.
Returns
-------
gain : (n_neurons,) array_like
Gain associated with each neuron. Sometimes denoted alpha.
bias : (n_neurons,) array_like
Bias current associated with each neuron.
"""
max_rates = np.array(max_rates, dtype=float, copy=False, ndmin=1)
intercepts = np.array(intercepts, dtype=float, copy=False, ndmin=1)
J_steps = 101 # Odd number so that 0 is a sample
max_rate = max_rates.max()
# Start with dummy gain and bias so x == J in rate calculation
gain = np.ones(J_steps)
bias = np.zeros(J_steps)
rate = np.zeros(J_steps)
# Find range of J that will achieve max rates (assume monotonic)
J_threshold = None
J_max = None
Jr = 10
for _ in range(10):
J = np.linspace(-Jr, Jr, J_steps)
rate = self.rates(J, gain, bias)
if J_threshold is None and (rate <= 0).any():
J_threshold = J[np.where(rate <= 0)[0][-1]]
if J_max is None and (rate >= max_rate).any():
J_max = J[np.where(rate >= max_rate)[0][0]]
if J_threshold is not None and J_max is not None:
break
else:
Jr *= 2
else:
if J_threshold is None:
raise RuntimeError("Could not find firing threshold")
if J_max is None:
raise RuntimeError("Could not find max current")
J = np.linspace(J_threshold, J_max, J_steps)
rate = self.rates(J, gain, bias)
gain = np.zeros_like(max_rates)
bias = np.zeros_like(max_rates)
J_tops = np.interp(max_rates, rate, J)
gain[:] = (J_threshold - J_tops) / (intercepts - 1)
bias[:] = J_tops - gain
return gain, bias
[docs] def max_rates_intercepts(self, gain, bias):
"""Compute the max_rates and intercepts given gain and bias.
Note that this default implementation is very slow! Whenever possible,
subclasses should override this with a neuron-specific implementation.
Parameters
----------
gain : (n_neurons,) array_like
Gain associated with each neuron. Sometimes denoted alpha.
bias : (n_neurons,) array_like
Bias current associated with each neuron.
Returns
-------
max_rates : (n_neurons,) array_like
Maximum firing rates of neurons.
intercepts : (n_neurons,) array_like
X-intercepts of neurons.
"""
max_rates = self.rates(np.ones_like(gain), gain, bias)
x_range = np.linspace(-1, 1, 101)
rates = np.asarray([self.rates(np.ones_like(gain) * x, gain, bias)
for x in x_range])
last_zeros = np.maximum(np.argmax(rates > 0, axis=0) - 1, 0)
intercepts = x_range[last_zeros]
return max_rates, intercepts
[docs] def rates(self, x, gain, bias):
"""Compute firing rates (in Hz) for given vector input, ``x``.
This default implementation takes the naive approach of running the
step function for a second. This should suffice for most rate-based
neuron types; for spiking neurons it will likely fail (those models
should override this function).
Parameters
----------
x : (n_neurons,) array_like
Vector-space input.
gain : (n_neurons,) array_like
Gains associated with each neuron.
bias : (n_neurons,) array_like
Bias current associated with each neuron.
Returns
-------
rates : (n_neurons,) ndarray
The firing rates at each given value of `x`.
"""
J = self.current(x, gain, bias)
out = np.zeros_like(J)
self.step_math(dt=1., J=J, output=out)
return out
[docs] def step_math(self, dt, J, output):
"""Implements the differential equation for this neuron type.
At a minimum, NeuronType subclasses must implement this method.
That implementation should modify the ``output`` parameter rather
than returning anything, for efficiency reasons.
Parameters
----------
dt : float
Simulation timestep.
J : (n_neurons,) array_like
Input currents associated with each neuron.
output : (n_neurons,) array_like
Output activities associated with each neuron.
"""
raise NotImplementedError("Neurons must provide step_math")
[docs]class Direct(NeuronType):
"""Signifies that an ensemble should simulate in direct mode.
In direct mode, the ensemble represents and transforms signals perfectly,
rather than through a neural approximation. Note that direct mode ensembles
with recurrent connections can easily diverge; most other neuron types will
instead saturate at a certain high firing rate.
"""
[docs] def gain_bias(self, max_rates, intercepts):
"""Always returns ``None, None``."""
return None, None
[docs] def max_rates_intercepts(self, gain, bias):
"""Always returns ``None, None``."""
return None, None
[docs] def rates(self, x, gain, bias):
"""Always returns ``x``."""
return np.array(x, dtype=float, copy=False, ndmin=1)
[docs] def step_math(self, dt, J, output):
"""Raises an error if called.
Rather than calling this function, the simulator will detect that
the ensemble is in direct mode, and bypass the neural approximation.
"""
raise SimulationError("Direct mode neurons shouldn't be simulated.")
# TODO: class BasisFunctions or Population or Express;
# uses non-neural basis functions to emulate neuron saturation,
# but still simulate very fast
[docs]class RectifiedLinear(NeuronType):
"""A rectified linear neuron model.
Each neuron is modeled as a rectified line. That is, the neuron's activity
scales linearly with current, unless it passes below zero, at which point
the neural activity will stay at zero.
Parameters
----------
amplitude : float
Scaling factor on the neuron output. Corresponds to the relative
amplitude of the output of the neuron.
"""
probeable = ('rates',)
def __init__(self, amplitude=1):
super(RectifiedLinear, self).__init__()
self.amplitude = amplitude
[docs] def gain_bias(self, max_rates, intercepts):
"""Determine gain and bias by shifting and scaling the lines."""
max_rates = np.array(max_rates, dtype=float, copy=False, ndmin=1)
intercepts = np.array(intercepts, dtype=float, copy=False, ndmin=1)
gain = max_rates / (1 - intercepts)
bias = -intercepts * gain
return gain, bias
[docs] def max_rates_intercepts(self, gain, bias):
"""Compute the inverse of gain_bias."""
intercepts = -bias / gain
max_rates = gain * (1 - intercepts)
return max_rates, intercepts
[docs] def step_math(self, dt, J, output):
"""Implement the rectification nonlinearity."""
output[...] = self.amplitude * np.maximum(0., J)
[docs]class SpikingRectifiedLinear(RectifiedLinear):
"""A rectified integrate and fire neuron model.
Each neuron is modeled as a rectified line. That is, the neuron's activity
scales linearly with current, unless the current is less than zero, at
which point the neural activity will stay at zero. This is a spiking
version of the RectifiedLinear neuron model.
Parameters
----------
amplitude : float
Scaling factor on the neuron output. Corresponds to the relative
amplitude of the output spikes of the neuron.
"""
probeable = ('spikes', 'voltage')
[docs] def rates(self, x, gain, bias):
"""Use RectifiedLinear to determine rates."""
J = self.current(x, gain, bias)
out = np.zeros_like(J)
RectifiedLinear.step_math(self, dt=1., J=J, output=out)
return out
[docs] def step_math(self, dt, J, spiked, voltage):
"""Implement the integrate and fire nonlinearity."""
voltage += np.maximum(J, 0) * dt
n_spikes = np.floor(voltage)
spiked[:] = self.amplitude * n_spikes / dt
voltage -= n_spikes
[docs]class Sigmoid(NeuronType):
"""A neuron model whose response curve is a sigmoid.
Since the tuning curves are strictly positive, the ``intercepts``
correspond to the inflection point of each sigmoid. That is,
``f(intercept) = 0.5`` where ``f`` is the pure sigmoid function.
"""
probeable = ('rates',)
tau_ref = NumberParam('tau_ref', low=0)
def __init__(self, tau_ref=0.0025):
super(Sigmoid, self).__init__()
self.tau_ref = tau_ref
@property
def _argreprs(self):
return [] if self.tau_ref == 0.0025 else ["tau_ref=%s" % self.tau_ref]
[docs] def gain_bias(self, max_rates, intercepts):
"""Analytically determine gain, bias."""
max_rates = np.array(max_rates, dtype=float, copy=False, ndmin=1)
intercepts = np.array(intercepts, dtype=float, copy=False, ndmin=1)
lim = 1. / self.tau_ref
inverse = -np.log(lim / max_rates - 1.)
gain = inverse / (1. - intercepts)
bias = inverse - gain
return gain, bias
[docs] def max_rates_intercepts(self, gain, bias):
"""Compute the inverse of gain_bias."""
inverse = gain + bias
intercepts = 1 - inverse / gain
lim = 1. / self.tau_ref
max_rates = lim / (1 + np.exp(-inverse))
return max_rates, intercepts
[docs] def step_math(self, dt, J, output):
"""Implement the sigmoid nonlinearity."""
output[...] = (1. / self.tau_ref) / (1.0 + np.exp(-J))
[docs]class LIFRate(NeuronType):
"""Non-spiking version of the leaky integrate-and-fire (LIF) neuron model.
Parameters
----------
tau_rc : float
Membrane RC time constant, in seconds. Affects how quickly the membrane
voltage decays to zero in the absence of input (larger = slower decay).
tau_ref : float
Absolute refractory period, in seconds. This is how long the
membrane voltage is held at zero after a spike.
amplitude : float
Scaling factor on the neuron output. Corresponds to the relative
amplitude of the output spikes of the neuron.
"""
probeable = ('rates',)
tau_rc = NumberParam('tau_rc', low=0, low_open=True)
tau_ref = NumberParam('tau_ref', low=0)
amplitude = NumberParam('amplitude', low=0, low_open=True)
def __init__(self, tau_rc=0.02, tau_ref=0.002, amplitude=1):
super(LIFRate, self).__init__()
self.tau_rc = tau_rc
self.tau_ref = tau_ref
self.amplitude = amplitude
@property
def _argreprs(self):
args = []
if self.tau_rc != 0.02:
args.append("tau_rc=%s" % self.tau_rc)
if self.tau_ref != 0.002:
args.append("tau_ref=%s" % self.tau_ref)
return args
[docs] def gain_bias(self, max_rates, intercepts):
"""Analytically determine gain, bias."""
max_rates = np.array(max_rates, dtype=float, copy=False, ndmin=1)
intercepts = np.array(intercepts, dtype=float, copy=False, ndmin=1)
inv_tau_ref = 1. / self.tau_ref if self.tau_ref > 0 else np.inf
if np.any(max_rates > inv_tau_ref):
raise ValidationError("Max rates must be below the inverse "
"refractory period (%0.3f)" % inv_tau_ref,
attr='max_rates', obj=self)
x = 1.0 / (1 - np.exp(
(self.tau_ref - (1.0 / max_rates)) / self.tau_rc))
gain = (1 - x) / (intercepts - 1.0)
bias = 1 - gain * intercepts
return gain, bias
[docs] def max_rates_intercepts(self, gain, bias):
"""Compute the inverse of gain_bias."""
intercepts = (1 - bias) / gain
max_rates = 1.0 / (self.tau_ref - self.tau_rc * np.log1p(
1.0 / (gain * (intercepts - 1) - 1)))
if not np.all(np.isfinite(max_rates)):
warnings.warn("Non-finite values detected in `max_rates`; this "
"probably means that `gain` was too small.")
return max_rates, intercepts
[docs] def rates(self, x, gain, bias):
"""Always use LIFRate to determine rates."""
J = self.current(x, gain, bias)
out = np.zeros_like(J)
# Use LIFRate's step_math explicitly to ensure rate approximation
LIFRate.step_math(self, dt=1, J=J, output=out)
return out
[docs] def step_math(self, dt, J, output):
"""Implement the LIFRate nonlinearity."""
j = J - 1
output[:] = 0 # faster than output[j <= 0] = 0
output[j > 0] = self.amplitude / (
self.tau_ref + self.tau_rc * np.log1p(1. / j[j > 0]))
# the above line is designed to throw an error if any j is nan
# (nan > 0 -> error), and not pass x < -1 to log1p
[docs]class LIF(LIFRate):
"""Spiking version of the leaky integrate-and-fire (LIF) neuron model.
Parameters
----------
tau_rc : float
Membrane RC time constant, in seconds. Affects how quickly the membrane
voltage decays to zero in the absence of input (larger = slower decay).
tau_ref : float
Absolute refractory period, in seconds. This is how long the
membrane voltage is held at zero after a spike.
min_voltage : float
Minimum value for the membrane voltage. If ``-np.inf``, the voltage
is never clipped.
amplitude : float
Scaling factor on the neuron output. Corresponds to the relative
amplitude of the output spikes of the neuron.
"""
probeable = ('spikes', 'voltage', 'refractory_time')
min_voltage = NumberParam('min_voltage', high=0)
def __init__(self, tau_rc=0.02, tau_ref=0.002, min_voltage=0, amplitude=1):
super(LIF, self).__init__(
tau_rc=tau_rc, tau_ref=tau_ref, amplitude=amplitude)
self.min_voltage = min_voltage
def step_math(self, dt, J, spiked, voltage, refractory_time):
# reduce all refractory times by dt
refractory_time -= dt
# compute effective dt for each neuron, based on remaining time.
# note that refractory times that have completed midway into this
# timestep will be given a partial timestep, and moreover these will
# be subtracted to zero at the next timestep (or reset by a spike)
delta_t = (dt - refractory_time).clip(0, dt)
# update voltage using discretized lowpass filter
# since v(t) = v(0) + (J - v(0))*(1 - exp(-t/tau)) assuming
# J is constant over the interval [t, t + dt)
voltage -= (J - voltage) * np.expm1(-delta_t / self.tau_rc)
# determine which neurons spiked (set them to 1/dt, else 0)
spiked_mask = voltage > 1
spiked[:] = spiked_mask * (self.amplitude / dt)
# set v(0) = 1 and solve for t to compute the spike time
t_spike = dt + self.tau_rc * np.log1p(
-(voltage[spiked_mask] - 1) / (J[spiked_mask] - 1))
# set spiked voltages to zero, refractory times to tau_ref, and
# rectify negative voltages to a floor of min_voltage
voltage[voltage < self.min_voltage] = self.min_voltage
voltage[spiked_mask] = 0
refractory_time[spiked_mask] = self.tau_ref + t_spike
[docs]class AdaptiveLIFRate(LIFRate):
"""Adaptive non-spiking version of the LIF neuron model.
Works as the LIF model, except with adapation state ``n``, which is
subtracted from the input current. Its dynamics are::
tau_n dn/dt = -n
where ``n`` is incremented by ``inc_n`` when the neuron spikes.
Parameters
----------
tau_n : float
Adaptation time constant. Affects how quickly the adaptation state
decays to zero in the absence of spikes (larger = slower decay).
inc_n : float
Adaptation increment. How much the adaptation state is increased after
each spike.
tau_rc : float
Membrane RC time constant, in seconds. Affects how quickly the membrane
voltage decays to zero in the absence of input (larger = slower decay).
tau_ref : float
Absolute refractory period, in seconds. This is how long the
membrane voltage is held at zero after a spike.
References
----------
.. [1] Koch, Christof. Biophysics of Computation: Information Processing
in Single Neurons. Oxford University Press, 1999. p. 339
"""
probeable = ('rates', 'adaptation')
tau_n = NumberParam('tau_n', low=0, low_open=True)
inc_n = NumberParam('inc_n', low=0)
def __init__(self, tau_n=1, inc_n=0.01, **lif_args):
super(AdaptiveLIFRate, self).__init__(**lif_args)
self.tau_n = tau_n
self.inc_n = inc_n
@property
def _argreprs(self):
args = super(AdaptiveLIFRate, self)._argreprs
if self.tau_n != 1:
args.append("tau_n=%s" % self.tau_n)
if self.inc_n != 0.01:
args.append("inc_n=%s" % self.inc_n)
return args
[docs] def step_math(self, dt, J, output, adaptation):
"""Implement the AdaptiveLIFRate nonlinearity."""
n = adaptation
LIFRate.step_math(self, dt, J - n, output)
n += (dt / self.tau_n) * (self.inc_n * output - n)
[docs]class AdaptiveLIF(AdaptiveLIFRate, LIF):
"""Adaptive spiking version of the LIF neuron model.
Works as the LIF model, except with adapation state ``n``, which is
subtracted from the input current. Its dynamics are::
tau_n dn/dt = -n
where ``n`` is incremented by ``inc_n`` when the neuron spikes.
Parameters
----------
tau_n : float
Adaptation time constant. Affects how quickly the adaptation state
decays to zero in the absence of spikes (larger = slower decay).
inc_n : float
Adaptation increment. How much the adaptation state is increased after
each spike.
tau_rc : float
Membrane RC time constant, in seconds. Affects how quickly the membrane
voltage decays to zero in the absence of input (larger = slower decay).
tau_ref : float
Absolute refractory period, in seconds. This is how long the
membrane voltage is held at zero after a spike.
References
----------
.. [1] Koch, Christof. Biophysics of Computation: Information Processing
in Single Neurons. Oxford University Press, 1999. p. 339
"""
probeable = ('spikes', 'adaptation', 'voltage', 'refractory_time')
[docs] def step_math(self, dt, J, output, voltage, ref, adaptation):
"""Implement the AdaptiveLIF nonlinearity."""
n = adaptation
LIF.step_math(self, dt, J - n, output, voltage, ref)
n += (dt / self.tau_n) * (self.inc_n * output - n)
[docs]class Izhikevich(NeuronType):
"""Izhikevich neuron model.
This implementation is based on the original paper [1]_;
however, we rename some variables for clarity.
What was originally 'v' we term 'voltage', which represents the membrane
potential of each neuron. What was originally 'u' we term 'recovery',
which represents membrane recovery, "which accounts for the activation
of K+ ionic currents and inactivation of Na+ ionic currents."
The 'a', 'b', 'c', and 'd' parameters are also renamed
(see the parameters below).
We use default values that correspond to regular spiking ('RS') neurons.
For other classes of neurons, set the parameters as follows.
* Intrinsically bursting (IB): ``reset_voltage=-55, reset_recovery=4``
* Chattering (CH): ``reset_voltage=-50, reset_recovery=2``
* Fast spiking (FS): ``tau_recovery=0.1``
* Low-threshold spiking (LTS): ``coupling=0.25``
* Resonator (RZ): ``tau_recovery=0.1, coupling=0.26``
Parameters
----------
tau_recovery : float, optional (Default: 0.02)
(Originally 'a') Time scale of the recovery variable.
coupling : float, optional (Default: 0.2)
(Originally 'b') How sensitive recovery is to subthreshold
fluctuations of voltage.
reset_voltage : float, optional (Default: -65.)
(Originally 'c') The voltage to reset to after a spike, in millivolts.
reset_recovery : float, optional (Default: 8.)
(Originally 'd') The recovery value to reset to after a spike.
References
----------
.. [1] E. M. Izhikevich, "Simple model of spiking neurons."
IEEE Transactions on Neural Networks, vol. 14, no. 6, pp. 1569-1572.
(http://www.izhikevich.org/publications/spikes.pdf)
"""
probeable = ('spikes', 'voltage', 'recovery')
tau_recovery = NumberParam('tau_recovery', low=0, low_open=True)
coupling = NumberParam('coupling', low=0)
reset_voltage = NumberParam('reset_voltage')
reset_recovery = NumberParam('reset_recovery')
def __init__(self, tau_recovery=0.02, coupling=0.2,
reset_voltage=-65., reset_recovery=8.):
super(Izhikevich, self).__init__()
self.tau_recovery = tau_recovery
self.coupling = coupling
self.reset_voltage = reset_voltage
self.reset_recovery = reset_recovery
@property
def _argreprs(self):
args = []
def add(attr, default):
if getattr(self, attr) != default:
args.append("%s=%s" % (attr, getattr(self, attr)))
add("tau_recovery", 0.02)
add("coupling", 0.2)
add("reset_voltage", -65.)
add("reset_recovery", 8.)
return args
[docs] def rates(self, x, gain, bias):
"""Estimates steady-state firing rate given gain and bias.
Uses the `.settled_firingrate` helper function.
"""
J = self.current(x, gain, bias)
voltage = np.zeros_like(J)
recovery = np.zeros_like(J)
return settled_firingrate(self.step_math, J, [voltage, recovery],
settle_time=0.001, sim_time=1.0)
[docs] def step_math(self, dt, J, spiked, voltage, recovery):
"""Implement the Izhikevich nonlinearity."""
# Numerical instability occurs for very low inputs.
# We'll clip them be greater than some value that was chosen by
# looking at the simulations for many parameter sets.
# A more principled minimum value would be better.
J = np.maximum(-30., J)
dV = (0.04 * voltage ** 2 + 5 * voltage + 140 - recovery + J) * 1000
voltage[:] += dV * dt
# We check for spikes and reset the voltage here rather than after,
# which differs from the original implementation by Izhikevich.
# However, calculating recovery for voltage values greater than
# threshold can cause the system to blow up, which we want
# to avoid at all costs.
spiked[:] = (voltage >= 30) / dt
voltage[spiked > 0] = self.reset_voltage
dU = (self.tau_recovery * (self.coupling * voltage - recovery)) * 1000
recovery[:] += dU * dt
recovery[spiked > 0] = recovery[spiked > 0] + self.reset_recovery
class NeuronTypeParam(Parameter):
def coerce(self, instance, neurons):
self.check_type(instance, neurons, NeuronType)
return super(NeuronTypeParam, self).coerce(instance, neurons)