Note
Go to the end to download the full example code.
Tutorial on learning to accumulate evidence with e-prop after Bellec et al. (2020)¶
Run this example as a Jupyter notebook:
See our guide for more information and troubleshooting.
Training a classification model using supervised e-prop plasticity to accumulate evidence.
Description¶
This script demonstrates supervised learning of a classification task with the eligibility propagation (e-prop) plasticity mechanism by Bellec et al. [1].
This type of learning is demonstrated at the proof-of-concept task in [1]. We based this script on their TensorFlow script given in [2].
The task, a so-called evidence accumulation task, is inspired by behavioral tasks, where a lab animal (e.g., a mouse) runs along a track, gets cues on the left and right, and has to decide at the end of the track between taking a left and a right turn of which one is correct. After a number of iterations, the animal is able to infer the underlying rationale of the task. Here, the solution is to turn to the side in which more cues were presented.
Learning in the neural network model is achieved by optimizing the connection weights with e-prop plasticity. This plasticity rule requires a specific network architecture depicted in Figure 1. The neural network model consists of a recurrent network that receives input from spike generators and projects onto two readout neurons - one for the left and one for the right turn at the end. The input neuron population consists of four groups: one group providing background noise of a specific rate for some base activity throughout the experiment, one group providing the input spikes of the left cues and one group providing them for the right cues, and a last group defining the recall window, in which the network has to decide. The readout neuron compares the network signal \(\pi_k\) with the teacher target signal \(\pi_k^*\), which it receives from a rate generator. Since the decision is at the end and all the cues are relevant, the network has to keep the cues in memory. Additional adaptive neurons in the network enable this memory. The network’s training error is assessed by employing a cross-entropy error loss.
Details on the event-based NEST implementation of e-prop can be found in [3].
References¶
Import libraries¶
We begin by importing all libraries required for the simulation, analysis, and visualization.
import matplotlib as mpl
import matplotlib.pyplot as plt
import nest
import numpy as np
from cycler import cycler
from IPython.display import Image
Schematic of network architecture¶
This figure, identical to the one in the description, shows the required network architecture in the center, the input and output of the classification task above, and lists of the required NEST device, neuron, and synapse models below. The connections that must be established are numbered 1 to 7.
try:
Image(filename="./eprop_supervised_classification_evidence-accumulation_bsshslm_2020.png")
except Exception:
pass
Setup¶
Initialize random generator¶
We seed the numpy random generator, which will generate random initial weights as well as random input and output.
rng_seed = 1 # numpy random seed
np.random.seed(rng_seed) # fix numpy random seed
Define timing of task¶
The task’s temporal structure is then defined, once as time steps and once as durations in milliseconds. Using a batch size larger than one aids the network in generalization, facilitating the solution to this task. The original number of iterations requires distributed computing. Increasing the number of iterations enhances learning performance up to the point where overfitting occurs. If early stopping is enabled, the classification error is tested in regular intervals and the training stopped as soon as the error selected as stop criterion is reached. After training, the performance can be tested over a number of test iterations.
batch_size = 32 # batch size, 64 in reference [2], 32 in the README to reference [2]
n_iter_train = 50 # number of training iterations, 2000 in reference [2]
n_iter_test = 4 # number of iterations for final test
do_early_stopping = True # if True, stop training as soon as stop criterion fulfilled
n_iter_validate_every = 10 # number of training iterations before validation
n_iter_early_stop = 8 # number of iterations to average over to evaluate early stopping condition
stop_crit = 0.07 # error value corresponding to stop criterion for early stopping
input = {
"n_symbols": 4, # number of input populations, e.g. 4 = left, right, recall, noise
"n_cues": 7, # number of cues given before decision
"prob_group": 0.3, # probability with which one input group is present
"spike_prob": 0.04, # spike probability of frozen input noise
}
steps = {
"cue": 100, # time steps in one cue presentation
"spacing": 50, # time steps of break between two cues
"bg_noise": 1050, # time steps of background noise
"recall": 150, # time steps of recall
}
steps["cues"] = input["n_cues"] * (steps["cue"] + steps["spacing"]) # time steps of all cues
steps["sequence"] = steps["cues"] + steps["bg_noise"] + steps["recall"] # time steps of one full sequence
steps["learning_window"] = steps["recall"] # time steps of window with non-zero learning signals
steps.update(
{
"offset_gen": 1, # offset since generator signals start from time step 1
"delay_in_rec": 1, # connection delay between input and recurrent neurons
"delay_rec_out": 1, # connection delay between recurrent and output neurons
"delay_out_norm": 1, # connection delay between output neurons for normalization
"extension_sim": 1, # extra time step to close right-open simulation time interval in Simulate()
"final_update": 3, # extra time steps to update all synapses at the end of task
}
)
steps["delays"] = steps["delay_in_rec"] + steps["delay_rec_out"] + steps["delay_out_norm"] # time steps of delays
steps["total_offset"] = steps["offset_gen"] + steps["delays"] # time steps of total offset
duration = {"step": 1.0} # ms, temporal resolution of the simulation
duration.update({key: value * duration["step"] for key, value in steps.items()}) # ms, durations
Set up simulation¶
As last step of the setup, we reset the NEST kernel to remove all existing NEST simulation settings and objects and set some NEST kernel parameters, some of which are e-prop-related.
params_setup = {
"eprop_learning_window": duration["learning_window"],
"eprop_reset_neurons_on_update": True, # if True, reset dynamic variables at start of each update interval
"eprop_update_interval": duration["sequence"], # ms, time interval for updating the synaptic weights
"print_time": False, # if True, print time progress bar during simulation, set False if run as code cell
"resolution": duration["step"],
"total_num_virtual_procs": 1, # number of virtual processes, set in case of distributed computing
}
nest.ResetKernel()
nest.set(**params_setup)
nest.set_verbosity("M_FATAL")
Create neurons¶
We proceed by creating a certain number of input, recurrent, and readout neurons and setting their parameters. Additionally, we already create an input spike generator and an output target rate generator, which we will configure later. Within the recurrent network, alongside a population of regular neurons, we introduce a population of adaptive neurons, to enhance the network’s memory retention.
n_in = 40 # number of input neurons
n_ad = 50 # number of adaptive neurons
n_reg = 50 # number of regular neurons
n_rec = n_ad + n_reg # number of recurrent neurons
n_out = 2 # number of readout neurons
params_nrn_out = {
"C_m": 1.0, # pF, membrane capacitance - takes effect only if neurons get current input (here not the case)
"E_L": 0.0, # mV, leak / resting membrane potential
"I_e": 0.0, # pA, external current input
"loss": "cross_entropy", # loss function
"regular_spike_arrival": False, # If True, input spikes arrive at end of time step, if False at beginning
"tau_m": 20.0, # ms, membrane time constant
"V_m": 0.0, # mV, initial value of the membrane voltage
}
params_nrn_reg = {
"beta": 1.0, # width scaling of the pseudo-derivative
"C_m": 1.0,
"c_reg": 300.0, # coefficient of firing rate regularization - 2*learning_window*(TF c_reg) for technical reasons
"E_L": 0.0,
"f_target": 10.0, # spikes/s, target firing rate for firing rate regularization
"gamma": 0.3, # height scaling of the pseudo-derivative
"I_e": 0.0,
"regular_spike_arrival": True,
"surrogate_gradient_function": "piecewise_linear", # surrogate gradient / pseudo-derivative function
"t_ref": 5.0, # ms, duration of refractory period
"tau_m": 20.0,
"V_m": 0.0,
"V_th": 0.6, # mV, spike threshold membrane voltage
}
# factors from the original pseudo-derivative definition are incorporated into the parameters
params_nrn_reg["gamma"] /= params_nrn_reg["V_th"]
params_nrn_reg["beta"] /= np.abs(params_nrn_reg["V_th"]) # prefactor is inside abs in the original definition
params_nrn_ad = {
"beta": 1.0,
"adapt_tau": 2000.0, # ms, time constant of adaptive threshold
"adaptation": 0.0, # initial value of the spike threshold adaptation
"C_m": 1.0,
"c_reg": 300.0,
"E_L": 0.0,
"f_target": 10.0,
"gamma": 0.3,
"I_e": 0.0,
"regular_spike_arrival": True,
"surrogate_gradient_function": "piecewise_linear",
"t_ref": 5.0,
"tau_m": 20.0,
"V_m": 0.0,
"V_th": 0.6,
}
params_nrn_ad["gamma"] /= params_nrn_ad["V_th"]
params_nrn_ad["beta"] /= np.abs(params_nrn_ad["V_th"])
params_nrn_ad["adapt_beta"] = 1.7 * (
(1.0 - np.exp(-duration["step"] / params_nrn_ad["adapt_tau"]))
/ (1.0 - np.exp(-duration["step"] / params_nrn_ad["tau_m"]))
) # prefactor of adaptive threshold
# Intermediate parrot neurons required between input spike generators and recurrent neurons,
# since devices cannot establish plastic synapses for technical reasons
gen_spk_in = nest.Create("spike_generator", n_in)
nrns_in = nest.Create("parrot_neuron", n_in)
# The suffix _bsshslm_2020 follows the NEST convention to indicate in the model name the paper
# that introduced it by the first letter of the authors' last names and the publication year.
nrns_reg = nest.Create("eprop_iaf_bsshslm_2020", n_reg, params_nrn_reg)
nrns_ad = nest.Create("eprop_iaf_adapt_bsshslm_2020", n_ad, params_nrn_ad)
nrns_out = nest.Create("eprop_readout_bsshslm_2020", n_out, params_nrn_out)
gen_rate_target = nest.Create("step_rate_generator", n_out)
nrns_rec = nrns_reg + nrns_ad
Create recorders¶
We also create recorders, which, while not required for the training, will allow us to track various dynamic variables of the neurons, spikes, and changes in synaptic weights. To save computing time and memory, the recorders, the recorded variables, neurons, and synapses can be limited to the ones relevant to the experiment, and the recording interval can be increased (see the documentation on the specific recorders). By default, recordings are stored in memory but can also be written to file.
n_record = 1 # number of neurons per type to record dynamic variables from - this script requires n_record >= 1
n_record_w = 5 # number of senders and targets to record weights from - this script requires n_record_w >=1
if n_record == 0 or n_record_w == 0:
raise ValueError("n_record and n_record_w >= 1 required")
params_mm_reg = {
"interval": duration["step"], # interval between two recorded time points
"record_from": ["V_m", "surrogate_gradient", "learning_signal"], # dynamic variables to record
"start": duration["offset_gen"] + duration["delay_in_rec"], # start time of recording
"label": "multimeter_reg",
}
params_mm_ad = {
"interval": duration["step"],
"record_from": params_mm_reg["record_from"] + ["V_th_adapt", "adaptation"],
"start": duration["offset_gen"] + duration["delay_in_rec"],
"label": "multimeter_ad",
}
params_mm_out = {
"interval": duration["step"],
"record_from": ["V_m", "readout_signal", "readout_signal_unnorm", "target_signal", "error_signal"],
"start": duration["total_offset"],
"label": "multimeter_out",
}
params_wr = {
"senders": nrns_in[:n_record_w] + nrns_rec[:n_record_w], # limit senders to subsample weights to record
"targets": nrns_rec[:n_record_w] + nrns_out, # limit targets to subsample weights to record from
"start": duration["total_offset"],
"label": "weight_recorder",
}
params_sr_in = {
"start": duration["offset_gen"],
"label": "spike_recorder_in",
}
params_sr_reg = {
"start": duration["offset_gen"],
"label": "spike_recorder_reg",
}
params_sr_ad = {
"start": duration["offset_gen"],
"label": "spike_recorder_ad",
}
mm_reg = nest.Create("multimeter", params_mm_reg)
mm_ad = nest.Create("multimeter", params_mm_ad)
mm_out = nest.Create("multimeter", params_mm_out)
sr_in = nest.Create("spike_recorder", params_sr_in)
sr_reg = nest.Create("spike_recorder", params_sr_reg)
sr_ad = nest.Create("spike_recorder", params_sr_ad)
wr = nest.Create("weight_recorder", params_wr)
nrns_reg_record = nrns_reg[:n_record]
nrns_ad_record = nrns_ad[:n_record]
Create connections¶
Now, we define the connectivity and set up the synaptic parameters, with the synaptic weights drawn from normal distributions. After these preparations, we establish the enumerated connections of the core network, as well as additional connections to the recorders.
params_conn_all_to_all = {"rule": "all_to_all", "allow_autapses": False}
params_conn_one_to_one = {"rule": "one_to_one"}
def calculate_glorot_dist(fan_in, fan_out):
glorot_scale = 1.0 / max(1.0, (fan_in + fan_out) / 2.0)
glorot_limit = np.sqrt(3.0 * glorot_scale)
glorot_distribution = np.random.uniform(low=-glorot_limit, high=glorot_limit, size=(fan_in, fan_out))
return glorot_distribution
dtype_weights = np.float32 # data type of weights - for reproducing TF results set to np.float32
weights_in_rec = np.array(np.random.randn(n_in, n_rec).T / np.sqrt(n_in), dtype=dtype_weights)
weights_rec_rec = np.array(np.random.randn(n_rec, n_rec).T / np.sqrt(n_rec), dtype=dtype_weights)
np.fill_diagonal(weights_rec_rec, 0.0) # since no autapses set corresponding weights to zero
weights_rec_out = np.array(calculate_glorot_dist(n_rec, n_out).T, dtype=dtype_weights)
weights_out_rec = np.array(np.random.randn(n_rec, n_out), dtype=dtype_weights)
params_common_syn_eprop = {
"optimizer": {
"type": "adam", # algorithm to optimize the weights
"batch_size": batch_size,
"beta_1": 0.9, # exponential decay rate for 1st moment estimate of Adam optimizer
"beta_2": 0.999, # exponential decay rate for 2nd moment raw estimate of Adam optimizer
"epsilon": 1e-8, # small numerical stabilization constant of Adam optimizer
"Wmin": -100.0, # pA, minimal limit of the synaptic weights
"Wmax": 100.0, # pA, maximal limit of the synaptic weights
},
"average_gradient": True, # if True, average the gradient over the learning window
"weight_recorder": wr,
}
eta_test = 0.0 # learning rate for test phase
eta_train = 5e-3 # learning rate for training phase
params_syn_base = {
"synapse_model": "eprop_synapse_bsshslm_2020",
"delay": duration["step"], # ms, dendritic delay
"tau_m_readout": params_nrn_out["tau_m"], # ms, for technical reasons pass readout neuron membrane time constant
}
params_syn_in = params_syn_base.copy()
params_syn_in["weight"] = weights_in_rec # pA, initial values for the synaptic weights
params_syn_rec = params_syn_base.copy()
params_syn_rec["weight"] = weights_rec_rec
params_syn_out = params_syn_base.copy()
params_syn_out["weight"] = weights_rec_out
params_syn_feedback = {
"synapse_model": "eprop_learning_signal_connection_bsshslm_2020",
"delay": duration["step"],
"weight": weights_out_rec,
}
params_syn_out_out = {
"synapse_model": "rate_connection_delayed",
"delay": duration["step"],
"receptor_type": 1, # receptor type of readout neuron to receive other readout neuron's signals for softmax
"weight": 1.0, # pA, weight 1.0 required for correct softmax computation for technical reasons
}
params_syn_rate_target = {
"synapse_model": "rate_connection_delayed",
"delay": duration["step"],
"receptor_type": 2, # receptor type over which readout neuron receives target signal
}
params_syn_static = {
"synapse_model": "static_synapse",
"delay": duration["step"],
}
params_init_optimizer = {
"optimizer": {
"m": 0.0, # initial 1st moment estimate m of Adam optimizer
"v": 0.0, # initial 2nd moment raw estimate v of Adam optimizer
}
}
nest.SetDefaults("eprop_synapse_bsshslm_2020", params_common_syn_eprop)
nest.Connect(gen_spk_in, nrns_in, params_conn_one_to_one, params_syn_static) # connection 1
nest.Connect(nrns_in, nrns_rec, params_conn_all_to_all, params_syn_in) # connection 2
nest.Connect(nrns_rec, nrns_rec, params_conn_all_to_all, params_syn_rec) # connection 3
nest.Connect(nrns_rec, nrns_out, params_conn_all_to_all, params_syn_out) # connection 4
nest.Connect(nrns_out, nrns_rec, params_conn_all_to_all, params_syn_feedback) # connection 5
nest.Connect(gen_rate_target, nrns_out, params_conn_one_to_one, params_syn_rate_target) # connection 6
nest.Connect(nrns_out, nrns_out, params_conn_all_to_all, params_syn_out_out) # connection 7
nest.Connect(nrns_in, sr_in, params_conn_all_to_all, params_syn_static)
nest.Connect(nrns_reg, sr_reg, params_conn_all_to_all, params_syn_static)
nest.Connect(nrns_ad, sr_ad, params_conn_all_to_all, params_syn_static)
nest.Connect(mm_reg, nrns_reg_record, params_conn_all_to_all, params_syn_static)
nest.Connect(mm_ad, nrns_ad_record, params_conn_all_to_all, params_syn_static)
nest.Connect(mm_out, nrns_out, params_conn_all_to_all, params_syn_static)
# After creating the connections, we can individually initialize the optimizer's
# dynamic variables for single synapses (here exemplarily for two connections).
nest.GetConnections(nrns_rec[0], nrns_rec[1:3]).set([params_init_optimizer] * 2)
Create input and output¶
We generate the input as four neuron populations, two producing the left and right cues, respectively, one the recall signal and one the background input throughout the task. The sequence of cues is drawn with a probability that favors one side. For each such sequence, the favored side, the solution or target, is assigned randomly to the left or right.
def generate_evidence_accumulation_input_output(batch_size, n_in, steps, input):
n_pop_nrn = n_in // input["n_symbols"]
prob_choices = np.array([input["prob_group"], 1 - input["prob_group"]], dtype=np.float32)
idx = np.random.choice([0, 1], batch_size)
probs = np.zeros((batch_size, 2), dtype=np.float32)
probs[:, 0] = prob_choices[idx]
probs[:, 1] = prob_choices[1 - idx]
batched_cues = np.zeros((batch_size, input["n_cues"]), dtype=int)
for b_idx in range(batch_size):
batched_cues[b_idx, :] = np.random.choice([0, 1], input["n_cues"], p=probs[b_idx])
input_spike_probs = np.zeros((batch_size, steps["sequence"], n_in))
for b_idx in range(batch_size):
for c_idx in range(input["n_cues"]):
cue = batched_cues[b_idx, c_idx]
step_start = c_idx * (steps["cue"] + steps["spacing"]) + steps["spacing"]
step_stop = step_start + steps["cue"]
pop_nrn_start = cue * n_pop_nrn
pop_nrn_stop = pop_nrn_start + n_pop_nrn
input_spike_probs[b_idx, step_start:step_stop, pop_nrn_start:pop_nrn_stop] = input["spike_prob"]
input_spike_probs[:, -steps["recall"] :, 2 * n_pop_nrn : 3 * n_pop_nrn] = input["spike_prob"]
input_spike_probs[:, :, 3 * n_pop_nrn :] = input["spike_prob"] / 4.0
input_spike_bools = input_spike_probs > np.random.rand(input_spike_probs.size).reshape(input_spike_probs.shape)
input_spike_bools[:, 0, :] = 0 # remove spikes in 0th time step of every sequence for technical reasons
target_cues = np.zeros(batch_size, dtype=int)
target_cues[:] = np.sum(batched_cues, axis=1) > int(input["n_cues"] / 2)
return input_spike_bools, target_cues
def get_params_task_input_output(n_iter_interval):
iteration_offset = n_iter_interval * batch_size * duration["sequence"]
dtype_in_spks = np.float32 # data type of input spikes - for reproducing TF results set to np.float32
input_spike_bools, target_cues = generate_evidence_accumulation_input_output(batch_size, n_in, steps, input)
input_spike_bools_arr = np.array(input_spike_bools).reshape(batch_size * steps["sequence"], n_in)
timeline_task = (
np.arange(0.0, batch_size * duration["sequence"], duration["step"]) + iteration_offset + duration["offset_gen"]
)
params_gen_spk_in = [
{"spike_times": timeline_task[input_spike_bools_arr[:, nrn_in_idx]].astype(dtype_in_spks)}
for nrn_in_idx in range(n_in)
]
target_rate_changes = np.zeros((n_out, batch_size))
target_rate_changes[np.array(target_cues), np.arange(batch_size)] = 1
params_gen_rate_target = [
{
"amplitude_times": np.arange(0.0, batch_size * duration["sequence"], duration["sequence"])
+ iteration_offset
+ duration["total_offset"],
"amplitude_values": target_rate_changes[nrn_out_idx],
}
for nrn_out_idx in range(n_out)
]
return params_gen_spk_in, params_gen_rate_target
Force final update¶
Synapses only get active, that is, the correct weight update calculated and applied, when they transmit a spike. To still be able to read out the correct weights at the end of the simulation, we force spiking of the presynaptic neuron and thus an update of all synapses, including those that have not transmitted a spike in the last update interval, by sending a strong spike to all neurons that form the presynaptic side of an eprop synapse. This step is required purely for technical reasons.
gen_spk_final_update = nest.Create("spike_generator", 1)
nest.Connect(gen_spk_final_update, nrns_in + nrns_rec, "all_to_all", {"weight": 1000.0})
Read out pre-training weights¶
Before we begin training, we read out the initial weight matrices so that we can eventually compare them to the optimized weights.
def get_weights(pop_pre, pop_post):
conns = nest.GetConnections(pop_pre, pop_post).get(["source", "target", "weight"])
conns["senders"] = np.array(conns["source"]) - np.min(conns["source"])
conns["targets"] = np.array(conns["target"]) - np.min(conns["target"])
conns["weight_matrix"] = np.zeros((len(pop_post), len(pop_pre)))
conns["weight_matrix"][conns["targets"], conns["senders"]] = conns["weight"]
return conns
weights_pre_train = {
"in_rec": get_weights(nrns_in, nrns_rec),
"rec_rec": get_weights(nrns_rec, nrns_rec),
"rec_out": get_weights(nrns_rec, nrns_out),
}
Simulate and evaluate¶
We train the network by simulating for a number of training iterations with the set learning rate. If early stopping is turned on, we evaluate the network’s performance on the validation set in regular intervals and, if the error is below a certain threshold, we stop the training early. If the error is not below the threshold, we continue training until the end of the set number of iterations. Finally, we evaluate the network’s performance on the test set. Furthermore, we evaluate the network’s training error by calculating a loss - in this case, the cross-entropy error between the integrated recurrent network activity and the target rate.
class TrainingPipeline:
def __init__(self):
self.results_dict = {
"error": [],
"loss": [],
"iteration": [],
"label": [],
}
self.n_iter_sim = 0
self.phase_label_previous = ""
self.error = 0
self.k_iter = 0
self.early_stop = False
def evaluate(self):
events_mm_out = mm_out.get("events")
readout_signal = events_mm_out["readout_signal"] # corresponds to softmax
target_signal = events_mm_out["target_signal"]
senders = events_mm_out["senders"]
times = events_mm_out["times"]
cond1 = times > (self.n_iter_sim - 1) * batch_size * duration["sequence"] + duration["total_offset"]
cond2 = times <= self.n_iter_sim * batch_size * duration["sequence"] + duration["total_offset"]
idc = cond1 & cond2
readout_signal = np.array([readout_signal[idc][senders[idc] == i] for i in set(senders)])
target_signal = np.array([target_signal[idc][senders[idc] == i] for i in set(senders)])
readout_signal = readout_signal.reshape((n_out, 1, batch_size, steps["sequence"]))
target_signal = target_signal.reshape((n_out, 1, batch_size, steps["sequence"]))
readout_signal = readout_signal[:, :, :, -steps["learning_window"] :]
target_signal = target_signal[:, :, :, -steps["learning_window"] :]
loss = -np.mean(np.sum(target_signal * np.log(readout_signal), axis=0), axis=(1, 2))
y_prediction = np.argmax(np.mean(readout_signal, axis=3), axis=0)
y_target = np.argmax(np.mean(target_signal, axis=3), axis=0)
accuracy = np.mean((y_target == y_prediction), axis=1)
errors = 1.0 - accuracy
self.results_dict["iteration"].append(self.n_iter_sim)
self.results_dict["error"].extend(errors)
self.results_dict["loss"].extend(loss)
self.results_dict["label"].append(self.phase_label_previous)
self.error = errors[0]
def run_phase(self, phase_label, eta):
params_common_syn_eprop["optimizer"]["eta"] = eta
nest.SetDefaults("eprop_synapse_bsshslm_2020", params_common_syn_eprop)
params_gen_spk_in, params_gen_rate_target = get_params_task_input_output(self.n_iter_sim)
nest.SetStatus(gen_spk_in, params_gen_spk_in)
nest.SetStatus(gen_rate_target, params_gen_rate_target)
self.simulate("total_offset")
self.simulate("extension_sim")
if self.n_iter_sim > 0:
self.evaluate()
duration["sim"] = batch_size * duration["sequence"] - duration["total_offset"] - duration["extension_sim"]
self.simulate("sim")
self.n_iter_sim += 1
self.phase_label_previous = phase_label
def run_training(self):
self.run_phase("training", eta_train)
def run_validation(self):
if do_early_stopping and self.k_iter % n_iter_validate_every == 0:
self.run_phase("validation", eta_test)
def run_early_stopping(self):
if do_early_stopping and self.k_iter % n_iter_validate_every == 0:
if self.k_iter > 0 and self.error < stop_crit:
errors_early_stop = []
for _ in range(n_iter_early_stop):
self.run_phase("early-stopping", eta_test)
errors_early_stop.append(self.error)
self.early_stop = np.mean(errors_early_stop) < stop_crit
def run_test(self):
for _ in range(n_iter_test):
self.run_phase("test", eta_test)
def simulate(self, k):
nest.Simulate(duration[k])
def run(self):
while self.k_iter < n_iter_train and not self.early_stop:
self.run_validation()
self.run_early_stopping()
self.run_training()
self.k_iter += 1
self.run_test()
self.simulate("total_offset")
self.simulate("extension_sim")
self.evaluate()
duration["task"] = self.n_iter_sim * batch_size * duration["sequence"] + duration["total_offset"]
gen_spk_final_update.set({"spike_times": [duration["task"] + duration["extension_sim"] + 1]})
self.simulate("final_update")
def get_results(self):
for k, v in self.results_dict.items():
self.results_dict[k] = np.array(v)
return self.results_dict
training_pipeline = TrainingPipeline()
training_pipeline.run()
results_dict = training_pipeline.get_results()
n_iter_sim = training_pipeline.n_iter_sim
Read out post-training weights¶
After the training, we can read out the optimized final weights.
weights_post_train = {
"in_rec": get_weights(nrns_in, nrns_rec),
"rec_rec": get_weights(nrns_rec, nrns_rec),
"rec_out": get_weights(nrns_rec, nrns_out),
}
Read out recorders¶
We can also retrieve the recorded history of the dynamic variables and weights, as well as detected spikes.
events_mm_reg = mm_reg.get("events")
events_mm_ad = mm_ad.get("events")
events_mm_out = mm_out.get("events")
events_sr_in = sr_in.get("events")
events_sr_reg = sr_reg.get("events")
events_sr_ad = sr_ad.get("events")
events_wr = wr.get("events")
Plot results¶
Then, we plot a series of plots.
do_plotting = True # if True, plot the results
if not do_plotting:
exit()
colors = {
"blue": "#2854c5ff",
"red": "#e04b40ff",
"green": "#25aa2cff",
"gold": "#f9c643ff",
"white": "#ffffffff",
}
plt.rcParams.update(
{
"axes.spines.right": False,
"axes.spines.top": False,
"axes.prop_cycle": cycler(color=[colors[k] for k in ["blue", "red", "green", "gold"]]),
}
)
Plot learning performance¶
We begin with two plots visualizing the learning performance of the network: the loss and the error, both plotted against the iterations.
fig, axs = plt.subplots(2, 1, sharex=True)
fig.suptitle("Learning performance")
for color, label in zip(colors, set(results_dict["label"])):
idc = results_dict["label"] == label
axs[0].scatter(results_dict["iteration"][idc], results_dict["loss"][idc], label=label)
axs[1].scatter(results_dict["iteration"][idc], results_dict["error"][idc], label=label)
axs[0].set_ylabel(r"$\mathcal{L} = -\sum_{t,k} \pi_k^{*,t} \log \pi_k^t$")
axs[1].set_ylabel("error")
axs[-1].set_xlabel("iteration")
axs[-1].legend(bbox_to_anchor=(1.05, 0.5), loc="center left")
axs[-1].xaxis.get_major_locator().set_params(integer=True)
fig.tight_layout()
Plot spikes and dynamic variables¶
This plotting routine shows how to plot all of the recorded dynamic variables and spikes across time. We take one snapshot in the first iteration and one snapshot at the end.
def plot_recordable(ax, events, recordable, ylabel, xlims):
for sender in set(events["senders"]):
idc_sender = events["senders"] == sender
idc_times = (events["times"][idc_sender] > xlims[0]) & (events["times"][idc_sender] < xlims[1])
ax.plot(events["times"][idc_sender][idc_times], events[recordable][idc_sender][idc_times], lw=0.5)
ax.set_ylabel(ylabel)
margin = np.abs(np.max(events[recordable]) - np.min(events[recordable])) * 0.1
ax.set_ylim(np.min(events[recordable]) - margin, np.max(events[recordable]) + margin)
def plot_spikes(ax, events, ylabel, xlims):
idc_times = (events["times"] > xlims[0]) & (events["times"] < xlims[1])
senders_subset = events["senders"][idc_times]
times_subset = events["times"][idc_times]
ax.scatter(times_subset, senders_subset, s=0.1)
ax.set_ylabel(ylabel)
margin = np.abs(np.max(senders_subset) - np.min(senders_subset)) * 0.1
ax.set_ylim(np.min(senders_subset) - margin, np.max(senders_subset) + margin)
for title, xlims in zip(
["Dynamic variables before training", "Dynamic variables after training"],
[
(0, steps["sequence"]),
((n_iter_sim - 1) * batch_size * steps["sequence"], n_iter_sim * batch_size * steps["sequence"]),
],
):
fig, axs = plt.subplots(14, 1, sharex=True, figsize=(8, 14), gridspec_kw={"hspace": 0.4, "left": 0.2})
fig.suptitle(title)
plot_spikes(axs[0], events_sr_in, r"$z_i$" + "\n", xlims)
plot_spikes(axs[1], events_sr_reg, r"$z_j$" + "\n", xlims)
plot_recordable(axs[2], events_mm_reg, "V_m", r"$v_j$" + "\n(mV)", xlims)
plot_recordable(axs[3], events_mm_reg, "surrogate_gradient", r"$\psi_j$" + "\n", xlims)
plot_recordable(axs[4], events_mm_reg, "learning_signal", r"$L_j$" + "\n(pA)", xlims)
plot_spikes(axs[5], events_sr_ad, r"$z_j$" + "\n", xlims)
plot_recordable(axs[6], events_mm_ad, "V_m", r"$v_j$" + "\n(mV)", xlims)
plot_recordable(axs[7], events_mm_ad, "surrogate_gradient", r"$\psi_j$" + "\n", xlims)
plot_recordable(axs[8], events_mm_ad, "V_th_adapt", r"$A_j$" + "\n(mV)", xlims)
plot_recordable(axs[9], events_mm_ad, "learning_signal", r"$L_j$" + "\n(pA)", xlims)
plot_recordable(axs[10], events_mm_out, "V_m", r"$v_k$" + "\n(mV)", xlims)
plot_recordable(axs[11], events_mm_out, "target_signal", r"$\pi^*_k$" + "\n", xlims)
plot_recordable(axs[12], events_mm_out, "readout_signal", r"$\pi_k$" + "\n", xlims)
plot_recordable(axs[13], events_mm_out, "error_signal", r"$\pi_k-\pi^*_k$" + "\n", xlims)
axs[-1].set_xlabel(r"$t$ (ms)")
axs[-1].set_xlim(*xlims)
fig.align_ylabels()
Plot weight time courses¶
Similarly, we can plot the weight histories. Note that the weight recorder, attached to the synapses, works differently than the other recorders. Since synapses only get activated when they transmit a spike, the weight recorder only records the weight in those moments. That is why the first weight registrations do not start in the first time step and we add the initial weights manually.
def plot_weight_time_course(ax, events, nrns, label, ylabel):
sender_label, target_label = label.split("_")
nrns_senders = nrns[sender_label]
nrns_targets = nrns[target_label]
for sender in set(events_wr["senders"]):
for target in set(events_wr["targets"]):
if sender in nrns_senders and target in nrns_targets:
idc_syn = (events["senders"] == sender) & (events["targets"] == target)
if np.any(idc_syn):
idc_syn_pre = (weights_pre_train[label]["source"] == sender) & (
weights_pre_train[label]["target"] == target
)
times = np.concatenate([[0.0], events["times"][idc_syn]])
weights = np.concatenate(
[np.array(weights_pre_train[label]["weight"])[idc_syn_pre], events["weights"][idc_syn]]
)
ax.step(times, weights, c=colors["blue"])
ax.set_ylabel(ylabel)
ax.set_ylim(-0.6, 0.6)
fig, axs = plt.subplots(3, 1, sharex=True, figsize=(3, 4))
fig.suptitle("Weight time courses")
nrns = {
"in": nrns_in.tolist(),
"rec": nrns_rec.tolist(),
"out": nrns_out.tolist(),
}
plot_weight_time_course(axs[0], events_wr, nrns, "in_rec", r"$W_\text{in}$ (pA)")
plot_weight_time_course(axs[1], events_wr, nrns, "rec_rec", r"$W_\text{rec}$ (pA)")
plot_weight_time_course(axs[2], events_wr, nrns, "rec_out", r"$W_\text{out}$ (pA)")
axs[-1].set_xlabel(r"$t$ (ms)")
axs[-1].set_xlim(0, duration["task"])
fig.align_ylabels()
fig.tight_layout()
Plot weight matrices¶
If one is not interested in the time course of the weights, it is possible to read out only the initial and final weights, which requires less computing time and memory than the weight recorder approach. Here, we plot the corresponding weight matrices before and after the optimization.
cmap = mpl.colors.LinearSegmentedColormap.from_list(
"cmap", ((0.0, colors["blue"]), (0.5, colors["white"]), (1.0, colors["red"]))
)
fig, axs = plt.subplots(3, 2, sharex="col", sharey="row")
fig.suptitle("Weight matrices")
all_w_extrema = []
for k in weights_pre_train.keys():
w_pre = weights_pre_train[k]["weight"]
w_post = weights_post_train[k]["weight"]
all_w_extrema.append([np.min(w_pre), np.max(w_pre), np.min(w_post), np.max(w_post)])
args = {"cmap": cmap, "vmin": np.min(all_w_extrema), "vmax": np.max(all_w_extrema)}
for i, weights in zip([0, 1], [weights_pre_train, weights_post_train]):
axs[0, i].pcolormesh(weights["in_rec"]["weight_matrix"].T, **args)
axs[1, i].pcolormesh(weights["rec_rec"]["weight_matrix"], **args)
cmesh = axs[2, i].pcolormesh(weights["rec_out"]["weight_matrix"], **args)
axs[2, i].set_xlabel("recurrent\nneurons")
axs[0, 0].set_ylabel("input\nneurons")
axs[1, 0].set_ylabel("recurrent\nneurons")
axs[2, 0].set_ylabel("readout\nneurons")
fig.align_ylabels(axs[:, 0])
axs[0, 0].text(0.5, 1.1, "before training", transform=axs[0, 0].transAxes, ha="center")
axs[0, 1].text(0.5, 1.1, "after training", transform=axs[0, 1].transAxes, ha="center")
axs[2, 0].yaxis.get_major_locator().set_params(integer=True)
cbar = plt.colorbar(cmesh, cax=axs[1, 1].inset_axes([1.1, 0.2, 0.05, 0.8]), label="weight (pA)")
fig.tight_layout()
plt.show()