Solving for decoders by simulating the neurons over time

Update: This feature is now in nengolib>=0.4.2. This requires nengo>=2.5.0. The basic usage is:

nengo.Connection(..., solver=nengolib.Temporal())

The complete documentation (optional parameters, example, etc.) is currently hosted here:

(Minor note: the order of the parameters has been swapped from the example below.)

In some situations it can be more accurate to explicitly simulate a population over time, collect its spiking activity, filter it, and then solve the least-squares problem on this data (as opposed to using the static rate-mode approximation). For instance, when using adaptive neuron models (e.g., AdaptiveLIF or Izhikevich), or, more generally, whenever the filtered spikes might convey more information than their corresponding rates.

Currently (as of Nengo 2.5.0), this takes considerable effort. The “standard” approach is to construct the entire model with a fixed set of seeds, simulate it, collect the training data, do the optimization yourself, reconstruct the model with the same seeds, and finally embed the new weights using a custom node or solver (see for instance). This can be tedious, inefficient, error-prone, and typically produces code that’s less modular. In practice, I’ve found that this “friction” will often deter me from applying this optimization to my own model.

To this end, this post demonstrates how to extend Nengo’s builder to have it automatically solve for the decoders by simulating the neurons over time (and filtering with some chosen synapse). This is enabled by simply wrapping whichever “underlying solver” you want (e.g., LstsqL2) with a custom solver (TemporalSolver; see below) when you create the connection. For instance,

nengo.Connection(..., solver=TemporalSolver())

defaults to LstsqL2 optimization, over time, with a 5 ms lowpass filter. The underlying solver and synapse may be passed as arguments to the TemporalSolver constructor. This makes it easy to quickly iterate, by testing this technique on some connections, without having to restructure the rest of your code, or perform multiple builds! Note, this uses the eval_points and function as you should normally expect, but the order of these points now matters – and you will likely want to filter the output of the function, using the same synapse, if your desired function is with respect to the unfiltered values (see example at very bottom).

This is currently only compatible with Nengo 2.5.0 and has not been extensively tested. Due to the way this has been implemented, this may not work with other backends, or in conjunction with other extensions to Nengo’s reference backend.

Please let me know if you have any issues, or if you can think of a better approach. And feel free to use/modify/redistribute, however you like, with or without attribution:

import numpy as np

import nengo
from nengo.builder import Builder
from nengo.builder.neurons import SimNeurons
from nengo.builder.signal import SignalDict
from nengo.config import SupportDefaultsMixin
from nengo.connection import ConnectionSolverParam
from nengo.params import Default
from nengo.solvers import Solver, LstsqL2
from nengo.synapses import SynapseParam, Lowpass

class TemporalSolver(Solver, SupportDefaultsMixin):
    """Wraps a solver to simulate the neurons over time."""

    solver = ConnectionSolverParam(
        'solver', default=LstsqL2(), readonly=True)
    synapse = SynapseParam(
        'synapse', default=Lowpass(tau=0.005), readonly=True)

    def __init__(self, solver=Default, synapse=Default):
        # We can't use super here because we need the defaults mixin
        # in order to determine self.solver.weights.
        self.solver = solver
        self.synapse = synapse
        Solver.__init__(self, weights=self.solver.weights)

    def mul_encoders(self, *args, **kwargs):
        return self.solver.mul_encoders(*args, **kwargs)

    def __call__(self, A, Y, rng=None, E=None):  # nengo issue #1358
        return self.solver.__call__(A, Y, rng=rng, E=E)

def build_solver(model, solver, conn, rng, transform):
    # Unpack the relevant variables from the connection.
    assert isinstance(conn.pre_obj, nengo.Ensemble)
    ensemble = conn.pre_obj
    neurons = ensemble.neurons
    neuron_type = ensemble.neuron_type

    # Find the operator that simulates the neurons.
    # We do it this way (instead of using the step_math method)
    # because we don't know the number of state parameters or their shapes.
    ops = list(filter(
        lambda op: (isinstance(op, SimNeurons) and
                    op.J is model.sig[neurons]['in']),
    if not len(ops) == 1:
        raise RuntimeError("Expected exactly one operator for simulating "
                           "neurons (%s), found: %s" % (neurons, ops))
    op = ops[0]

    # Create stepper for the neuron model.
    signals = SignalDict()
    step_simneurons = op.make_step(signals, model.dt, rng)

    # Create custom rates method that uses the built neurons.
    def override_rates_method(x, gain, bias):
        n_eval_points, n_neurons = x.shape
        assert ensemble.n_neurons == n_neurons

        a = np.empty((n_eval_points, n_neurons))
        for i, x_t in enumerate(x):
            signals[op.J][...] = neuron_type.current(x_t, gain, bias)
            a[i, :] = signals[op.output]
        return solver.synapse.filt(a, axis=0, y0=0, dt=model.dt)

    # Hot-swap the rates method while calling the underlying solver.
    # The solver will then call this temporarily created rates method
    # to process each evaluation point.
    save_rates_method = neuron_type.rates
    neuron_type.rates = override_rates_method
        # Note: passing solver.solver doesn't actually cause solver.solver
        # to be built. It will still use conn.solver. The only point of
        # passing solver.solver is to invoke its corresponding builder
        # function (in case something custom happens to be registered).
        return, conn, rng, transform)
        neuron_type.rates = save_rates_method

Below is a demo application that highlights the potential performance gains of this approach, by building a communication channel out of Izhikevich neurons.

The TemporalSolver takes 3 seconds to build on my machine, while the default solver takes over 100 seconds due to the way it processes each evaluation point. The normalized error decreases from 0.644 to 0.458 – a 29 percent reduction in error.

The only difference between the two figures is that the “Temporal Solver” uses a process to determine the activities over time, while the “Default Solver” determines the rate in response to each of these points independently.

import matplotlib.pyplot as plt
from nengolib.signal import nrmse

def demo(n_neurons=100,
         neuron_type=nengo.Izhikevich(coupling=0.25),  # low-threshold spiking (LTS)
         train_process=nengo.processes.WhiteSignal(10.0, high=5, y0=0, rms=0.3, seed=0),
         test_process=nengo.processes.WhiteSignal(10.0, high=5, y0=0, rms=0.3, seed=1),
    """Filtered communication channel demo."""

    kwargs = {'eval_points': train_process.run_steps(n_eval_points, dt=dt, d=dimensions)}
    if temporal_solver:
            'solver': TemporalSolver(synapse=synapse),
            'function': synapse.filt(kwargs['eval_points'], axis=0, y0=0, dt=dt)})

    with nengo.Network() as model:
        stim = nengo.Node(output=test_process, size_out=dimensions)
        x = nengo.Ensemble(n_neurons, dimensions, neuron_type=neuron_type)
        out = nengo.Node(size_in=dimensions)

        nengo.Connection(stim, x, synapse=None)
        nengo.Connection(x, out, synapse=None, **kwargs)

        p = nengo.Probe(out, synapse=synapse)
        ideal = nengo.Probe(stim, synapse=synapse)

    with nengo.Simulator(model, dt=dt) as sim:

    plt.figure(figsize=(16, 5))
    plt.title(r"%s Solver + %d %s Neurons $\rightarrow$ NRMSE=%.3f" % (
        "Temporal" if temporal_solver else "Default",
        n_neurons, type(neuron_type).__name__,
    plt.plot(sim.trange(),[p], label="Actual", alpha=0.8)
    plt.plot(sim.trange(),[ideal], label="Ideal",
             linestyle='--', lw=3, alpha=0.8)
    plt.xlabel("Time (s)")

if __name__ == '__main__':


Awesome! I would recommend that the TemporalSolver class be added to nengo_extras, and the example put in the examples folder there.

And just from a quick glance, I think that this should work with backends that use the nengo builder, like nengo_ocl.


Very nice! I’ve been concerned about adaptive neurons also. Perhaps even for non-adaptive neurons, the temporal solver will turn out more accurate?

For time-varying input encoded with adaptive neurons, the decoding may not be accurate, as the present spiking depends on the previous input. Seems to me a limitation of the NEF framework? Any thoughts?

Great, thanks! I will do this after it’s tested a bit more.

Thanks for checking. Will be great if it just works!

You can try copy-pasting my code and changing the neuron_type to nengo.LIF(). In this case, the NRMSE actually increases from 0.058 to 0.060, but they are too close to tell. It’s also going to be difficult to distinguish these cases, for one, because the L2-regularization acts differently on the two matrices (there is filtered noise “built in” to one of the matrices).

For higher-frequency inputs I agree with you that I’d expect the TemporalSolver to become more accurate. However, when the input doesn’t change too quickly, my current understanding is that the standard NEF/Nengo approach is pretty near-optimal for non-adaptive neurons (in the feed-forward case). I am working on some unpublished analysis here.

My current opinion here is that this should be understood as a problem of how to specify what dynamics you (as the network designer) would like for your network. Adaptive neurons should allow for certain classes of dynamical systems to become more accurate with fewer neurons. The “real” problem is that it’s hard to characterize what these systems are in a useful way (at the level of vector/function representation). We have a recent paper on arXiv that lays some groundwork, but requires the designer to use an approximation such as linear-nonlinear modelling.

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Sorry for the delay in replying. Just noticed your recent work:
and got reminded to again look at the references you’d linked.

Awesome that you’re able to handle higher-order or delayed (but still only linear right?) synapses in the NEF. Also the linear-non-linear work by @Eric is cool. Yes, these two approaches together could handle adaptive neurons with even short term plastic synapses perhaps? Will write more once I explore further, esp. with regard to online local learning. Also the updated version of my arxiv pre-print, on an adaptive control theory based learning rule very similar to PES for the NEF, is now at:


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Ended up needing this again today. Is it okay if I make a PR in nengo-extras using this code?

On second thought I plan to put this into nengolib as this fits pretty naturally into my thesis. I have an open issue for this, but think the only important point to address is that synapse=None should be a valid option. This should be a simple enough change, but I will need some time to do the PR as it needs some unit testing.

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Finally got this into nengolib>=0.4.2. I’ve updated the very top of this thread.

Documentation is currently here: (scroll to the bottom for a complete example)

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