Relating a Spiking Neural Network Model and the Diffusion Model of Decision-Making

Abstract

Many models of decision-making assume accumulation of evidence to threshold as a core mechanism to predict response probabilities and response times. A spiking neural network model (Wang, 2002) instantiates these mechanisms at the level of biophysically-plausible pools of neurons with excitatory and inhibitory connections and has numerous model parameters tuned by physiological measures. The diffusion model (Ratcliff, 1978) is a cognitive model that can be fitted to a range of behaviors and conditions. We investigated how parameters of the cognitive-level diffusion model relate to the parameters of a neural-level spiking model. In each simulated “experiment,” we generated “data” from the spiking neural network by factorially combining a manipulation of choice difficulty (via the input to the spiking model) and a manipulation of one of the core parameters of the spiking model. We then fitted the diffusion model to these simulated data to observe how manipulation of each core spiking model parameter mapped on to fitted drift rate, response threshold, and non-decision time. Manipulations of parameters in the spiking model related to input sensitivity, threshold, and stimulus processing time mapped on to their conceptual analogues in the diffusion model, namely drift rate, threshold, and non-decision time. Manipulations of parameters in the spiking model with no direct analogue to the diffusion model, non-stimulus-specific background input, strength of recurrent excitation, and receptor conductances mapped on to threshold in the diffusion model. We discuss implications of these results for interpretations of fits of the diffusion model to behavioral data.

Appendix Spiking Neural Network Model Dynamics

Neuron and synapse models. Each neuron in the spiking network is modeled as a leaky integrate-and-fire neuron (e.g., Abbott, 1999) with a membrane potential V(t) defined by a differential equation

$${\mathrm C}_{\mathrm m}\frac{\mathrm{dV}(\mathrm t)}{\mathrm{dt}}=-{\mathrm g}_{\mathrm L}\left(\mathrm V\left(\mathrm t\right)-{\mathrm V}_{\mathrm L}\right)-{\mathrm I}_{\mathrm{syn}}(\mathrm t)$$

where C_m is the membrane capacitance, g_L is the leakage conductance, V_L is the resting potential, and I_syn(t) is the synaptic current. When the membrane potential V(t) of a neuron reaches a threshold potential V_th = -50 mV, the neuron generates a spike and is reset to V_r = − 55 mV for a refractory period of T_ref. For excitatory neurons, C_m = 0.5 nF, g_L = 25 nS, V_L = -70 mV, and T_ref = 2 ms; for inhibitory neurons, C_m = 0.2 nF, g_L = 20 nS, V_L = -70 mV, and T_ref = 1 ms. We fixed these parameters at values that replicate known biophysical properties of cortical neurons (as per Wang, 2002).

The total synaptic current I_syn(t) is the sum of external input currents and excitatory and inhibitory currents from recurrent network connections. There are four types of currents at the synaptic connections: external AMPA, recurrent AMPA, recurrent NMDA, and recurrent GABA. Therefore, the equation for total synaptic current I_syn(t) is:

$${\mathrm I}_{\mathrm{syn}}\left(\mathrm t\right)={\mathrm I}_{\mathrm{ext},\mathrm{AMPA}}(\mathrm t)+{\mathrm I}_{\mathrm{rec},\mathrm{AMPA}}(\mathrm t)+{\mathrm I}_{\mathrm{NMDA}}(\mathrm t)+{\mathrm I}_{\mathrm{GABA}}(\mathrm t)$$

The individual receptor currents are given by:

$${\mathrm I}_{\mathrm{ext},\mathrm{AMPA}}\left(\mathrm t\right)={\mathrm g}_{\mathrm{ext},\mathrm{AMPA}}\left(\mathrm V\left(\mathrm t\right)-{\mathrm V}_{\mathrm E}\right){\mathrm s}_{\mathrm{ext},\mathrm{AMPA}}(\mathrm t)$$

$${\mathrm I}_{\mathrm{rec},\mathrm{AMPA}}\left(\mathrm t\right)={\mathrm g}_{\mathrm{rec},\mathrm{AMPA}}\left(\mathrm V\left(\mathrm t\right)-{\mathrm V}_{\mathrm E}\right)\sum\limits_{\mathrm j=1}^{\mathrm{Ce}}{\mathrm w}_{\mathrm j}{\mathrm s}_{\mathrm j,\mathrm{AMPA}}(\mathrm t)$$

$${\mathrm I}_{\mathrm{rec},\mathrm{NMDA}}\left(\mathrm t\right)=\frac{{\mathrm g}_{\mathrm{NMDA}}\left(\mathrm V\left(\mathrm t\right)-{\mathrm V}_{\mathrm E}\right)}{1+\lbrack\mathrm{Mg}^{2+}\rbrack\mathrm e^{-0.062\mathrm V(\mathrm t)/3.57}}\sum\limits_{\mathrm j=1}^{\mathrm{Ce}}{\mathrm w}_{\mathrm j}{\mathrm s}_{\mathrm j,\mathrm{NMDA}}(t)$$

$${\mathrm I}_{\mathrm{rec},\mathrm{GABA}}\left(\mathrm t\right)={\mathrm g}_{\mathrm{GABA}}\left(\mathrm V\left(\mathrm t\right)-{\mathrm V}_{\mathrm I}\right)\sum\limits_{\mathrm j=1}^{\mathrm{Ci}}{\mathrm s}_{\mathrm j,\mathrm{GABA}}(\mathrm t)$$

where V_E = 0 and V_I = − 70 mV are the reversal potentials and [Mg²⁺] = 1 mM is the extracellular magnesium concentration. The sums are over all excitatory connections C_e or all inhibitory connections C_i. We fixed these parameters at values that replicate known biophysical properties of cortical neurons (as per Wang, 2002).

The g variables are conductance values of the receptor channels. All default conductance values were chosen to match known biophysical measurements (Wang, 2002). For excitatory neurons g_ext,AMPA = 2.100 nS, g_rec,AMPA = 0.050 nS, g_NMDA = 0.165 nS, and g_GABA = 1.300 nS. For inhibitory neurons g_ext,AMPA = 1.620 nS, g_rec,AMPA = 0.040 nS, g_NMDA = 0.130, and g_GABA = 1.000 nS. To explore the relationship between diffusion model parameters and conductance values, in simulated experiments, we used the following ranges: g_rec,AMPA = 0.045, 0.0475, 0.050 (default), 0.0525, and 0.055 nS; g_NMDA = 0.163, 0.164, 0.165 (default), 0.166, and 0.167 nS; and g_GABA = 1.290, 1.295, 1.300 (default), 1.305, and 1.310 nS. As with other choices of manipulated parameter values, these values were chosen because they allowed the model to exhibit appropriate competitive dynamics while also producing reasonable differences in predicted behavior.

The w variables are synaptic weights that control the strength of connections between neural pools. The network is connected all-to-all, meaning that a single neuron, j, is connected to every other neuron in the network with weight w_j. The synaptic weights were structured with a “Hebbian rule” with stronger connections between neurons representing the same choice (w_j = w₊) and weaker connections between neurons representing opposite choices (w_j = w_–). All other connections assume w_j = 1. Following Wang (2002), we used the default value w₊ = 1.7 and w_– was determined by the Eq. 1 – f(w₊ – 1)/(1 – f). f is the proportion of excitatory neurons in the spiking neural network that belong to a choice-selective subpool (in our case, f = 240/1600 = 0.15). To identify the relationship between diffusion model parameters and synaptic weights, we explored the following ranges: w₊ = 1.650, 1.675, 1.700 (default), 1.725, and 1.750. Beyond these ranges, the model failed to exhibit competitive dynamics (Wong & Wang, 2006).

The s variables are gating variables and represent the fraction of receptor channels that are open at a given time. These parameters control the receptor dynamics and are presumed to be a fixed biophysical property of the cells. They are governed by the following equations:

$$\frac{{\mathrm{ds}}_{\mathrm j,\mathrm{AMPA}}(\mathrm t)}{\mathrm{dt}}=\sum_{\mathrm k}\delta(\mathrm t-{\mathrm t}_{\mathrm j}^{\mathrm k})-\frac{{\mathrm s}_{\mathrm j,\mathrm{AMPA}}(\mathrm t)}{{\mathrm\tau}_{\mathrm{AMPA}}}$$

where τ_AMPA = 2 ms is the decay time and the sum is over the presynaptic spikes from neuron j. For external AMPA currents, the sum is over Poisson spike trains generated with means specified by the parameters μ_ext, μ₁, and μ₂ that are independent for each cell. For NMDA receptors:

$$\frac{{\mathrm{ds}}_{\mathrm j,\mathrm{NMDA}}(\mathrm t)}{\mathrm{dt}}=\alpha{\mathrm x}_{\mathrm j}(\mathrm t)(1-{\mathrm s}_{\mathrm j,\mathrm{NMDA}}\left(\mathrm t\right))-\frac{{\mathrm s}_{\mathrm j,\mathrm{NMDA}}(\mathrm t)}{{\mathrm\tau}_{\mathrm{NMDA},\mathrm{decay}}}$$

$$\frac{{\mathrm{dx}}_{\mathrm j}(\mathrm t)}{\mathrm{dt}}=\sum_{\mathrm k}\delta(\mathrm t-{\mathrm t}_{\mathrm j}^{\mathrm k})-\frac{{\mathrm x}_{\mathrm j}(\mathrm t)}{{\mathrm\tau}_{\mathrm{NMDA},\mathrm{rise}}}$$

where τ_NMDA,decay = 100 ms and α = 0.5/ms. The rise time dynamics of NMDA receptors is modeled by x_j, where τ_NMDA,rise = 2 ms. Finally, for GABA:

$$\frac{{\mathrm{ds}}_{\mathrm j,\mathrm{GABA}}(\mathrm t)}{\mathrm{dt}}=\sum_{\mathrm k}\delta(\mathrm t-{\mathrm t}_{\mathrm j}^{\mathrm k})-\frac{{\mathrm s}_{\mathrm j,\mathrm{GABA}}(\mathrm t)}{{\mathrm\tau}_{\mathrm{GABA}}}$$

where τ_GABA = 100 ms.