Sensitivity to perturbation

This script simulates a network in two successive trials, which are identical except for one extra input spike in the second realisation. (a small perturbation). The network consists of recurrent, randomly connected excitatory and inhibitory neurons. Its activity is driven by an external Poisson input provided to all neurons independently. In order to ensure that the network is reset appropriately between the trials, we do the following steps: - resetting the network - resetting the random network generator - resetting the internal clock - deleting all entries in the spike detector - introducing a hyperpolarisation phase between the trials (in order to avoid that spikes remaining in the NEST memory after the first simulation are fed into the second simulation)

Importing all necessary modules for simulation, analysis and plotting.

import numpy
import pylab
import nest

Here we define all parameters necessary for building and simulating the network. We start with the global network parameters.

NE = 1000      # number of excitatory neurons
NI = 250       # number of inhibitory neurons
N = NE + NI    # total number of neurons
KE = 100       # excitatory in-degree
KI = 25        # inhibitory in-degree

Parameters specific for the neurons in the network. The default values of the reset potential "E_L" and the spiking threshold "V_th" are used to set the limits of the initial potential of the neurons.

neuron_model = 'iaf_psc_delta'
neuron_params = nest.GetDefaults(neuron_model)
Vmin = neuron_params['E_L']   # minimum of initial potential distribution (mV)
Vmax = neuron_params['V_th']  # maximum of initial potential distribution (mV)

Synapse parameters. Changing the weights (J) in the network can lead to qualitatively different behaviours. If J is small (e.g. J = 0.1) we are likely to observe a non-chaotic network behaviour (after perturbation the network returns to its original activity). Increasing J (e.g J = 5.5) leads to rather chaotic activity. Given that in this example the transition to chaos is probabilistic, we sometimes observe chaotic behaviour for small weights (e.g. J = 0.5) and non-chaotic behaviour for strong weights (e.g. J = 5.4).

J = 0.5                   # excitatory synaptic weight (mV)
g = 6.                    # relative inhibitory weight
delay = 0.1               # spike transmission delay (ms)

External input parameters.

Jext = 0.2                # PSP amplitude for external Poisson input (mV)
rate_ext = 6500.          # rate of the external Poisson input

Perturbation parameters.

t_stim = 400.             # perturbation time (time of the extra spike)
Jstim = Jext              # perturbation amplitude (mV)

Simulation parameters.

T = 1000.                 # simulation time per trial (ms)
fade_out = 2.*delay       # fade out time (ms)
dt = 0.01                 # simulation time resolution (ms)
seed_NEST = 30            # seed of random number generator in Nest
seed_numpy = 30           # seed of random number generator in numpy

Before we build the network we reset the simulation Kernel (to make sure that previous NEST simulations in the python shell will not disturb this simulation), and set the simulation resolution (later defined synaptic delays cannot be smaller then the simulation resolution).

nest.SetStatus([0], [{"resolution": dt}])

Now we start building the network and create excitatory and inhibitory nodes and connect them. According to the connectivity specification each neurons gets KE (KI) synapses from the excitatory (inhibitory) population assigned randomly.

nodes_ex = nest.Create(neuron_model, NE)
nodes_in = nest.Create(neuron_model, NI)
allnodes = nodes_ex+nodes_in

nest.Connect(nodes_ex, allnodes,
             conn_spec={'rule': 'fixed_indegree', 'indegree': KE},
             syn_spec={'weight': J, 'delay': dt})
nest.Connect(nodes_in, allnodes,
             conn_spec={'rule': 'fixed_indegree', 'indegree': KI},
             syn_spec={'weight': -g*J, 'delay': dt})

Afterwards we create a Poisson generator that provides spikes (the external input) to the neurons until time 'T' is reached. Afterwards a DC generator, which is also connected to the whole population, provides a stong hyperpolarisation step for a short time period 'fade_out'. In order to suppress the firing after the simulation successfully the fade out period has to last at least two times the simulation resolution.

ext = nest.Create("poisson_generator",
                  params={'rate': rate_ext, 'stop': T})
nest.Connect(ext, allnodes,
             syn_spec={'weight': Jext, 'delay': dt})

suppr = nest.Create("dc_generator",
                    params={'amplitude': -1e16, 'start': T,
                            'stop': T+fade_out})
nest.Connect(suppr, allnodes)

spikedetector = nest.Create("spike_detector")
nest.Connect(allnodes, spikedetector)

Creating the spike generator that provides the extra spike (perturbation).

stimulus = nest.Create("spike_generator")
nest.SetStatus(stimulus, {'spike_times': []})

Finally we run the two simulations successively. After each simulation the sender ids and spiketimes are stored in a list ('senders', 'spiketimes').

senders = []
spiketimes = []

for trial in [0, 1]:

As mentioned above, we need to reset the network, the random number generator, and the clock of the simulation Kernel. In addition we make sure that there is no spike left in the spike detector.

    nest.SetStatus([0], [{"rng_seeds": [seed_NEST]}])
    nest.SetStatus([0], {'time': 0.0})
    nest.SetStatus(spikedetector, {'n_events': 0})

We assign random initial membrane potentials to all neurons (uniform disbributed between Vmin and Vthreshold)

    Vms = Vmin + (Vmax - Vmin) * numpy.random.rand(N)
    nest.SetStatus(allnodes, "V_m", Vms)

In the second trial we add an extra input spike at time t_stim to the neuron that fires first after perturbation time t_stim. Thus, we make sure that the perturbation is transmitted to the network before it fades away in the perturbed neuron. (Single IAF-neurons are not chaotic.)

    if trial == 1:
        id_stim = [senders[0][spiketimes[0] > t_stim][0]]
        nest.Connect(stimulus, list(id_stim),
                     syn_spec={'weight': Jstim, 'delay': dt})
        nest.SetStatus(stimulus, {'spike_times': [t_stim]})

Now we simulate the network and add a fade out period to get rid of the remaining spikes in the current NEST memory (see introduction).


Storing the data.

    senders += [nest.GetStatus(spikedetector, 'events')[0]['senders']]
    spiketimes += [nest.GetStatus(spikedetector, 'events')[0]['times']]

Plotting the spiking activity of the network (first trial in red, second trial in black).

pylab.plot(spiketimes[0], senders[0], 'ro', ms=4.)
pylab.plot(spiketimes[1], senders[1], 'ko', ms=2.)
pylab.xlabel('time (ms)')
pylab.ylabel('neuron id')
pylab.xlim((0, T))
pylab.ylim((0, N))