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Impairments in decision-making are frequently observed in neurodegenerative diseases, but the mechanisms underlying such pathologies remain elusive. In this work, we study, on the basis of novel time-delayed neuronal population model, if the delay in self-inhibition terms can explain those impairments. Analysis of proposed system reveals that there can be up to three positive steady states, with the one having the lowest neuronal activity being always locally stable in nondelayed case. We show, however, that this steady state becomes unstable above a critical delay value for which, in certain parameter ranges, a subcritical Hopf bifurcation occurs. We then apply psychometric function to translate model-predicted ring rates into probabilities that a decision is being made. Using numerical simulations, we demonstrate that for small synaptic delays the decision-making process depends directly on the strength of supplied stimulus and the system correctly identifies to which population the stimulus was applied. However, for delays above the Hopf bifurcation threshold we observe complex impairments in the decision-making process; that is, increasing the strength of the stimulus may lead to the change in the neuronal decision into a wrong one. Furthermore, above critical delay threshold, the system exhibits ambiguity in the decision-making.

Gamma-Aminobutyric Acid (GABA) is the most prevalent inhibitory neurotransmitter in the human brain [

On the other hand, in the aging process, the sensory capacity is declining, which affects the cognitive functions [

In 1996, Salthouse [

In this work, we propose a mechanism linking the two aspects of aging in cortical networks: the neurodegeneration in the local inhibitory synapses and the processing-speed related impairments in perceptual decision-making. This mechanism is based on a neuronal population model of decision-making based on a winner-take-all mechanism. The novelty lies in combining a winner-take-all mechanism well routed in the decision-making neuroscience, with the system of delayed differential equations representing the local inhibition within the two competing populations. With the use of this model, we are able to demonstrate that, for small synaptic delays in the local inhibition within the competing populations, the decision-making process depends directly on the strength of the stimulus, and the network is able to correctly identify the direction the stimulus came from. However, large delays can lead to a subcritical Hopf bifurcation resulting in complex decision-making process impairments. In particular, we demonstrate that, above the Hopf bifurcation point, increasing the strength of the stimulus can confuse the network and cause a wrong decision to be made. Furthermore, for delay values above critical threshold the system exhibits ambiguity in the decision-making. This effect can explain how the experimentally found difficulties in decision-making in elderly adults can be caused by loss in cognitive capacities [

The paper is organized in the following way. In Section

The famous perceptual experiment on rhesus monkeys [

In this work, we focus on modeling the most basic perceptual decision-making in neuronal networks. In such conditions, the network needs to disambiguate between two sensory stimuli. We consider changes in the firing rates

A neuronal population model of decision-making. Neurons in the first population project to the neurons in the second population and vice versa. Both neuronal populations receive self-inhibition with delay (

In order to describe the temporal dynamics of considered system (Figure

In reality both populations considered in the proposed model are embedded in a larger network; therefore even in the absence of the population-specific sensory stimulus, they receive a constant input. Therefore, in the resting state, this system receives equal constant inputs

In this section, we provide the analytical results regarding the behavior of solutions of (

In this subsection, we present a detailed analysis of the model dynamics for

Let us consider

For

For

Notice that due to its symmetric structure the system described by (

It is obvious that the symmetry of (

Examples of the phase space portrait of (

Now, we aim to study the influence of the delay on the dynamics of the model. Our main goal is to show that there exists such time delay

Before starting the analysis, we first state some general results that will be useful in this section. Let us consider a general DDE

Assume that

We only need to check transversality condition. The derivative

Lemma

The next lemma is a simple consequence of Proposition 1 from [

Let

Following [

Now, we turn to the main topic of this subsection that is analysis of (

Let us denote a characteristic matrix by

As a result, we can state the following theorem.

Let

We only need to check that

For the reference values of parameters for

Hence, we study the bifurcation at

In the following, we base on the ideas presented in [

When looking for

Next, we need to find a vector

The type of the studied bifurcation is determined by a coefficient

If

First, we calculate the denominator of

Next, we would like to calculate the numerator of

Using the formula above, we calculate

Let us denote

Now, we are in a position to check the sign of

Notice that, for small

Now, we calculate

In order to complement the analysis above, we can also show that in the reference case the second bifurcation appearing for

Next, we calculate

At the end we sum up the results of the Hopf bifurcation analysis in the following corollary.

For reference parameter values the system described by (

In order to simulate the decision-making process, we assume a transient change in one of the inputs to the nodes; that is, we start from the resting state of the network and then solve (

In all the numerical experiments, we have chosen

One important aspect of the considered (

We start our numerical experiments from simulations in which stimulus of a fixed magnitude

Comparison of the model solutions ((a); (

Interestingly, for delays above the critical value at which first Hopf bifurcation occurs, we observe very nonintuitive behavior; see middle panels in Figures

Because of the observed nonintuitive behavior of the psychometric function, we decided to calculate a decision map; that is, to evaluate the psychometric function values on the model solutions for different values of delay

(a) Decision map resulting from the psychometric function evaluation on solutions to (

In Figure

Exemplary solution of (

In this study, we model perceptual decision-making in elderly individuals. Although there is an extensive evidence that the cognitive impairments in aging relate to the slow processing of information [

In order to bridge this gap, we propose to study the influence of synaptic delays on making perceptual choices with use of a population model of decision-making based on a winner-take-all mechanism. In this simple model, the network needs to make a binary choice between the two options. The inspiration for this model was a spiking neuronal network model with a slow reverberation mechanism by Wang [

We achieve two main results that can contribute to the understanding of the associations between the delayed GABA-signaling and the cognitive impairments during aging. Firstly, the decision-making performance is dependent on the synaptic delays in the local inhibition. For the delays below the critical value, we observe a clear association between the magnitude of the stimulus and the probability of making the correct decision (Figure

The second important result from our study is that, for delay values above critical threshold, the system exhibits ambiguity in decision-making, reflected by decision switches. This result can account for the slow reaction times in perceptual decision-making in the elderly [

Our model predicts that impairment in the local inhibition in the cortex can result in the impaired decision-making. Although GABA concentration in prefrontal cortex and perceptual decision-making are both affected by aging, there is a lack of computational models characterizing the causal link between the two. Therefore, the model should be validated in laboratory conditions. The prediction given by the model is hard to test in the human cohorts because recordings from interneurons in the cortex are invasive. However, there are now tools in translational psychiatry that make this validation viable. For instance, the decision-making quality can be evaluated in mice in multiple experimental paradigms [

One remark to make is that our model is qualitative rather than quantitative, and the same mechanism can be encountered at different time scales of inhibition and for different configurations of inputs and stimuli. There are multiple GABAergic receptors in the cortex, and they have their own characteristic timescales. In example, the fast mode of inhibition is related to the GABA-A receptors [

As a summary, the model proposed in this work yields new insights into the mechanisms of aging in cortical circuits, mediated by neurodegeneration in the local inhibitory synapses.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors developed the model as a team. Urszula Foryś conducted the derivations in a collaboration with Martyna Płomecka and Katarzyna Piskała. Jan Poleszczuk performed and visualized numerical simulations. Natalia Z. Bielczyk formulated the introduction and the discussion part of the manuscript.

The authors would like to thank Maciej Borodzik, Piotr Kozakowski, and Franciszek Rakowski for the discussion and advice. The research leading to these results has received funding from the University of Warsaw, the European Community’s Seventh Framework Programme (FP7/2007–2013) under Grant Agreement no. 305697 (OPTIMISTIC), the European Community’s Seventh Framework Programme (FP7/2007–2013) under Grant Agreement no. 278948 (TACTICS), European Union’s Seventh Framework Programme for Research, Technological Development and Demonstration under Grant Agreement no. 603016 (MATRICS), and National Science Center in Poland under Grant Agreement no. 2011/03/N/ST1/00109.

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