Deep Learning for Parameter Recovery from a Neural Mass Model of Perceptual Decision-Making

In neuroscience, parameter recovery refers to the problem of ﬁnding the best parameters of a model for ﬁt-ting the experimental data. The developing of more biologically plausible computational models of cognition has offered a signiﬁcant improvement in the predictive power at the cost of a higher complexity posing increasing challenges on parameter recovery. Here, we present a deep learning approach to recover parameters of a two-variables neural mass model simulating evidence accumulation during perceptual decision-making. We show that our algorithm is able to recovery well speciﬁc set of parameters but might fail when trying to predict combinations of parameters with a high degree of interaction, i.e. parameters that have inherently similar effects on the model’s output. Thus, our study suggests that deep learning for parameter recovery should go together with a carefully designed experiment to study the effects of different parameters that are not richly interacting.


Introduction
In cognitive neuroscience, an increasing attention has been put into biologically more realistic models able to capture the cellular and circuit mechanisms instead of relying only on more abstract behavioral models.In general, these models are more complex than behavioral ones meaning that they may have a higher number of parameters and more complex differential equations.Therefore, although they fit the behavioral data very well, it is not straightforward to recover their parameters.Recent works have tackled the problem of parameter recovery from non-differentiable behavioral decision making models like the diffusion model for conflict tasks (White, Servant, & Logan, 2018) or the leaky competing accumulator (Miletić, Turner, Forstmann, & van Maanen, 2017).However, the problem of parameter recovery of biologically realistic models of decision making remains under-represented in literature.In this study, we perform parameter recovery of a neural model using deep learning as it is more computationally efficient and reliable than other global optimization procedures (Stonkute, 2019).

Methods
We simulated the data using the two competing variables neural mass model designed by Wong and Wang (2006) which reproduces the evidence accumulation involved in decisionmaking during a random dot leftward or rightward motion discrimination task.The two populations interact via mutual inhibitory connections and self excitatory mechanisms.The input is modeled as two different synaptic currents generated by the stimulus and it is stronger at higher dot movement coherence level.The model also considers a common non-selective background input and a noisy current.Here, we used an extension of the model where the common background current is modulated by a factor β to take into account the speedaccuracy trade-off (Standage, Wang, & Blohm, 2014;Heitz & Schall, 2012).The equations describing the output synaptic currents of the two populations l and r are: where I l,r are the stimulus-generated input currents, r l,r are the firing rates of the neurons of the two populations, S l,r are the gating variables representing the activity of the two populations and I noise,l,r Gaussian noise currents.These variables are described by differential equations which are not reported for simplicity, see (Wong & Wang, 2006) for a full description of the model.For the parameters, J ll = J rr = 0.2609nA is the self-excitatory coupling strength, J lr = J rl = 0.0497nA the mutual inhibitory coupling strength and β is the modulation of the background current I 0 = 0.3255nA.
We focused on recover four parameters: the excitatory and inhibitory coupling strengths J ll and J lr , the background current modulation β and we defined a new parameter τ nd called non-decision time as a simple constant of time added to the reaction time for each decision.Initially, the parameter values have been drawn from the following uniform distributions: J ll = J rr ∈ [0.25, 0.30]nA; J lr = J rl ∈ [0, 0.25]nA; β ∈ [0.95, 1.05]; τ nd ∈ [100, 500]ms.
To determine the intervals we searched for parameter values that gave a biologically plausible outputs.For example, we expected the accuracy to be around 0.5 at 0% coherence and increasing following a sigmoid-like function to saturate at 1 for higher coherence levels (Wong & Wang, 2006;Roitman & Shadlen, 2002).
For each set of parameters, we computed the equations for a total of 10,000 trials.Each trial had a maximum duration of 2.5s for a total of 2,500 iterations of 1ms each.A decision is made if the activity S l,r of one of the two accumulators reached a threshold set at 15Hz.For simplicity, we assumed a positive value for the coherence meaning that the correct direction of 384 This work is licensed under the Creative Commons Attribution 3.0 Unported License.To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0 the points was always leftward.The output for each trial was then 0 for rightward/wrong decisions, or 1 for leftward/correct decisions.If the trial ended without either accumulators activities reaching the threshold, an output of −1 indicating a nondecision has been generated.If the non-decision rate was higher than 0.2 we discarded the sample.The results were used to compute a set of features describing the performance of the model for each set of parameters.We used the following 13 features: the accuracy calculated as the fraction of correct choices over the total choices made, average reaction times for correct and wrong choices and the 10th, 30th, 50th, 70th and 90th percentiles of the reaction times distributions for wrong and correct choices.
We performed training and hyperparameter optimization on a deep neural network with three fully connected layers using the 13 features as input and the four parameters as outputs (all the Python scripts for data sampling and deep learning training are publicly available at the repository https://github.com/Emalude/wwparrecdl.

Results
The model fails to predict J ll values (R 2 = 0.11) and performs slightly better for the other parameters with R 2 scores of 0.86, 0.69 and 0.88 for J lr , β and τ nd respectively.We hypothesized that the difficulty of prediction might be due to the concurrency of J ll and β with a similar influence on the outputs of the model.To test this hypothesis, we generated two new batches of samples; in the first, we kept J ll = 0.2609 constant.In the second we set β = 1.We repeated hyperparameter tuning and training separately for the two batches with three parameters as output.The results with the absence of modulation β are shown in Figure 1.The R 2 scores are 0.95, 0.99 and 0.99 for J ll , J lr and τ nd respectively.When J ll is constant, R 2 scores are 0.94, 0.90 and 0.92 for J lr , β and τ nd respectively.

Discussion
Our results suggest that parameters with a similar influence on the output are difficult to recover with high accuracy.Therefore, it is important to carefully design an experiment and combine the parameter selection with previous knowledge about the effects of those parameters and their variability across subjects or conditions.For example, suppose that by analyzing the behavioral outputs of a random dot motion experiment on healthy and pathological subjects, our algorithm recovers a significantly lower self-excitatory synaptic strength in subjects with a certain pathology compared to healthy ones.We could not use this as a final proof of the fact that the pathology lowers J ll because we first need to address the possibility that such difference in the behavioral output is caused by other parameters having a similar effects on the outputs in their range of values (for example β).