ThresholdSAM

class spikeometric.models.ThresholdSAM(r: float, b: float, tau: float, dt: float, sigma: float, rho: float, theta: float, rng: torch._C.Generator)[source]

Bases: spikeometric.models.sa_model.SAModel

The generative model used in the paper “Systematic errors in connectivity inferred from activity in strongly coupled recurrent circuits”.

Is is a threshold-based model, where the neurons spike if the synaptic activation is above a threshold. Since the model is threshold-based, it is not tunable.

It is defined by the following equations:

  1. \[g_i(t+1) = r \: \sum_j (W_0)_{j,i}\: s_j(t) + b_i + \mathcal{E}_i(t+1)\]
  2. \[\begin{split}X_i(t+1) = \begin{cases} 1 & \text{if } g_i(t+1) > \theta \\ 0 & \text{otherwise} \end{cases}\end{split}\]

where \(r\) is the scaling of the recurrent connections, \(b_i\) is the strength of a background input and \(\mathcal{E}_i(t+1)\) is an external stimulus. The response to the input is then compared to a threshold \(\theta\) and the neurons spike if the activation is above the threshold.

Between each time step, the synaptic activation is decayed by a factor of \((1-\Delta t/\tau)\) if the neuron did not spike and increased by \(\Delta t\) if it did.

Parameters

  • r (float) – The scaling of the recurrent connections. (tunable)

  • b (float) – The strength of the background input. (tunable)

  • tau (float) – The time constant of the neurons.

  • dt (float) – The time step of the simulation.

  • sigma (float) – The standard deviation of the noise.

  • rho (float) – The sparsity of the noise.

  • theta (float) – The threshold of the neurons (the neurons will spike if the activation is above the threshold).

  • rng (torch.Generator) – The random number generator.

input(edge_index: torch.Tensor, W: torch.Tensor, state: torch.Tensor, t=- 1) → torch.Tensor[source]

Calculates the input to each neuron as:

\[g_i(t+1) = r \: \sum_j (W_0)_{j,i}\: s_j(t) + b_i + \mathcal{E}_i(t+1)\]
Parameters

  • edge_index (torch.Tensor [2, n_edges]) – The connectivity of the network.

  • W (torch.Tensor [n_edges, 1]) – The edge weights of the connectivity filter.

  • state (torch.Tensor [n_neurons, 1]) – The activation of the neurons at time t.

  • t (int) – The current time step.

Returns

input – The input to each neuron at time t.

Return type

torch.Tensor [n_neurons]

background_input(n_neurons: int)[source]

Generate the background input. This is a uniform excitatory input given by the parameter \(b\) plus a scaled Gaussian noise with standard deviation \(\sigma\) and some sparsity.

Parameters

  • t (int) – The current time step.

  • n_neurons (int) – The number of neurons in the network.

Returns

background_input

Return type

torch.Tensor [n_edges]

emit_spikes(input: torch.Tensor) → torch.Tensor[source]

Emit spikes from the network. The neuron will spike if the input it receives is above the threshold.

\[\begin{split}X_i(t+1) = \begin{cases} 1 & \text{if } g_i(t+1) > \theta \\ 0 & \text{otherwise} \end{cases}\end{split}\]
Parameters

activation (torch.Tensor [n_neurons]) – The activation of the neurons at the current time step.

Returns

spikes – The spikes of the network at the current time step.

Return type

torch.Tensor [n_neurons]

update_activation(spikes: torch.Tensor, activation: torch.Tensor) → torch.Tensor[source]

Update the activation of the neurons according to the formula:

\[s_i(t+1) = s_i(t)(1 - \frac{\Delta t}{\tau}) + X_i(t)\Delta t\]

where \(s_i(t)\) is the activation of neuron \(i\) at time \(t\), \(X_i(t)\) indicates whether the neuron spiked at time \(t\) or not, and \(\tau\) is the time constant of the neurons.

Parameters

  • spikes (torch.Tensor [n_neurons, 1]) – The spikes of the network at the current time step.

  • activation (torch.Tensor [n_neurons, 1]) – The activation of the neurons at the previous time step.

Returns

activation – The activation of the neurons at the current time step.

Return type

torch.Tensor [n_neurons, 1]