.. currentmodule:: brian2 .. Tetzlaff_2015: Example: Tetzlaff_2015 ====================== .. only:: html .. |launchbinder| image:: http://mybinder.org/badge.svg .. _launchbinder: https://mybinder.org/v2/gh/brian-team/brian2-binder/master?filepath=examples/frompapers/Tetzlaff_2015.ipynb .. note:: You can launch an interactive, editable version of this example without installing any local files using the Binder service (although note that at some times this may be slow or fail to open): |launchbinder|_ Reproduces Figure 2F of The Use of Hebbian Cell Assemblies for Nonlinear Computation by Tetzlaff C., Dasgupta S., Kulvicius T. and Wörgötter F. Sci Rep 5, 12866 (2015). https://doi.org/10.1038/srep12866 Sebastian Schmitt, 2022 :: import numpy as np import matplotlib.pyplot as plt from brian2 import NeuronGroup, Synapses, StateMonitor, run, defaultclock, ms, second, TimedArray, seed # random seed that gives curves similar to the ones in the publication seed(9873487) # neuron parameters (sigmoidal activation) beta = 0.03 epsilon = 120 F_max = 100 F_T = 1 tau_u = 1*ms R = 0.012 # plasticity timescales tau_ratio = 60 # hebbian tau_H = 3e4*ms # synaptic scaling tau_SS = tau_ratio * tau_H # synaptic weights W_max = np.sqrt(tau_ratio*(F_max**2/(F_max - F_T))) W_ext = W_max W_input = W_max W_I = 0.3*W_max # stimulus N_units = 100 N_stim_units = 20 stim_A_units_until = N_stim_units stim_B_units_from = N_units-N_stim_units # connection probabilities p_E = 0.1 p_I = 0.2 # paper uses 0.3*ms DT = 0.5*ms defaultclock.dt = DT # duration of a learning trial lt = 5000*DT duration = 100*lt no_input_until = 5*lt balanced_until = duration/2 # gate balanced presentation of stimulus 1 and 2 balanced = TimedArray([lt_counter*lt < balanced_until for lt_counter in range(int(duration/lt))], dt=lt) # function used for stimulus (typo in paper, +1 is not part of the argument of sin) stim_func = TimedArray([100*(np.sin(0.1*(i+1))+1) for i in range(int(duration/DT))], dt=DT) # gate learning phase of either stimulus 1 or 2 learning_phase = TimedArray([i%10 > 3 for i in range(int(duration/(0.1*lt)))], dt=0.1*lt) # if not balanced present stimulus A three times more often than stimulus B stim_A_gate = TimedArray([lt_counter % 2 == 0 if balanced(lt_counter*lt) else lt_counter % 4 in [0,1,2] for lt_counter in range(int(duration/lt))], dt=lt) stim_B_gate = TimedArray([lt_counter % 2 == 1 if balanced(lt_counter*lt) else lt_counter % 4 == 3 for lt_counter in range(int(duration/lt))], dt=lt) # noise is applied also during stimulation neurons = NeuronGroup(N_units, """ F = F_max/(1+exp(beta*(epsilon-u))) : 1 du/dt = (-u + R*(I_E - I_I + W_input*(I_stim_A + I_stim_B)))/tau_u + R*W_ext*20*sqrt((DT/ms)/ms)*xi: 1 I_E : 1 I_I : 1 index : 1 (constant) stim_units_A = index < stim_A_units_until : boolean stim_units_B = index >= (stim_B_units_from) : boolean I_stim_A = learning_phase(t)*int(stim_units_A)*stim_A_gate(t)*stim_func(t) : 1 I_stim_B = learning_phase(t)*int(stim_units_B)*stim_B_gate(t)*stim_func(t) : 1 """, method = "euler") neurons.index = range(len(neurons)) # excitatory connections with Hebbian plasticity and synaptic scaling synapses_E = Synapses(neurons, neurons, """ dw/dt = 1/tau_H*F_pre*F_post + 1/tau_SS*(F_T - F_post)*w**2 : 1 (clock-driven) I_E_post = w*F_pre : 1 (summed) """, method="euler" ) # do not connect between the two populations of stimulated neurons synapses_E.connect(p=p_E, condition="((j > stim_A_units_until and i >= stim_B_units_from) or (j < stim_B_units_from and i < stim_A_units_until))" "or ((i > stim_A_units_until and i < stim_B_units_from) and (j > stim_A_units_until and j < stim_B_units_from))") # fixed weight inhibitory connections synapses_I = Synapses(neurons, neurons, """ w : 1 I_I_post = w*F_pre : 1 (summed) """ ) synapses_I.connect(p=p_I) synapses_I.w = W_I statemon_neurons = StateMonitor(neurons, ["F", "I_stim_A", "I_stim_B"], record=True, dt=100*defaultclock.dt) statemon_synapses_E = StateMonitor(synapses_E, "w", record=True, dt=100*defaultclock.dt) statemon_synapses_for_assembly_analysis = StateMonitor(synapses_E, "w", record=True, dt=lt) run(duration, report="text") # threshold saying that synaptic efficacies larger than theta are # 'strong' and others are 'weak' theta = 0.5*W_max in_assembly_A = [] in_assembly_B = [] # traverse through the graph following 'strong' synapses def go(W, source, units_in_assembly): units_in_assembly.add(source) # check all possible targets for target in range(N_units): w = W[source][target] if w > theta: W[source][target] = 0 go(W, target, units_in_assembly) # for each learning trial for ws in statemon_synapses_for_assembly_analysis.w.T: # construct a full weight matrix W = np.full((N_units, N_units), np.nan) W[synapses_E.i[:], synapses_E.j[:]] = ws for in_assembly, stim_units in zip([in_assembly_A, in_assembly_B], [range(stim_A_units_until), range(stim_B_units_from, N_units)]): units_in_assembly = set() # start with units that are stimulated for stim_unit in stim_units: go(W, stim_unit, units_in_assembly) in_assembly.append(len(units_in_assembly)) # competitive development of the two competing cell assemblies A and B as a function of the input protocol fig, ax = plt.subplots() ax.plot(in_assembly_A, linestyle="None", marker='o', color='orange', label="A") ax.plot(in_assembly_B, linestyle="None", marker='o', color='olivedrab', label="B") ax.set_ylim(19, 51) ax.set_xlim(0, 100) ax.set_ylabel("Neurons in Cell Assembly [%]") ax.set_xlabel("Learning Trial") ax.axvline(balanced_until/lt, linestyle='dashed', color='k') ax.text(15, 52, " A A", color='orange', fontfamily="monospace", fontsize="xx-large") ax.text(15, 52, " B B", color='olivedrab', fontfamily="monospace", fontsize="xx-large") ax.text(65, 52, " 3A 3A", color='orange', fontfamily="monospace", fontsize="xx-large") ax.text(65, 52, " B B", color='olivedrab', fontfamily="monospace", fontsize="xx-large") plt.show() # stimulus, neuronal activity and excitatory weights as function of time fig, axes = plt.subplots(3, sharex=True) axes[0].plot(statemon_neurons.I_stim_A[0], label="A", color='orange') axes[0].plot(statemon_neurons.I_stim_B[-1], label="B", color='olivedrab') axes[0].legend(loc="upper right") axes[0].set_title("Stimulus") axes[1].imshow(statemon_neurons.F, aspect='auto') axes[1].set_title("Neuron Activity") axes[1].axhline(stim_A_units_until, linestyle='dashed', color='white') axes[1].axhline(stim_B_units_from, linestyle='dashed', color='white') axes[2].imshow(statemon_synapses_E.w, aspect='auto') axes[2].set_title("Excitatory Weights") axes[2].set_xticks(range(0, 5000, 250)) axes[2].set_xticklabels(f"{i}" for i in range(0, 100, 5)) axes[2].set_xlabel("Learning Trial") axes[2].set_xlim(0, 5000) fig.tight_layout() plt.show() .. image:: ../resources/examples_images/frompapers.Tetzlaff_2015.1.png .. image:: ../resources/examples_images/frompapers.Tetzlaff_2015.2.png