Neural mechanisms balancing accuracy and flexibility in working memory and decision tasks

Abstract

The living system follows the principles of physics, yet distinctive features, such as adaptability, differentiate it from conventional systems. The cognitive functions of decision-making (DM) and working memory (WM) are crucial for animal adaptation, but the underlying mechanisms are still unclear. To explore the mechanism underlying DM and WM functions, here we applied a general non-equilibrium landscape and flux approach to a biophysically based model that can perform decision-making and working memory functions. Our findings reveal that DM accuracy improved with stronger resting states in the circuit architecture with selective inhibition. However, the robustness of working memory against distractors was weakened. To address this, an additional non-selective input during the delay period of decision-making tasks was proposed as a mechanism to gate distractors with minimal increase in thermodynamic cost. This temporal gating mechanism, combined with the selective-inhibition circuit architecture, supports a dynamical modulation that emphasizes the robustness or flexibility to incoming stimuli in working memory tasks according to the cognitive task demands. Our approach offers a quantitative framework to uncover mechanisms underlying cognitive functions grounded in non-equilibrium physics.

Introduction

Living systems, composed of chemical substances and requiring energy, adhere to the principles of physics and chemistry. However, these systems possess unique attributes, such as self-organization, self-replication, and adaptability, that distinguish them from conventional physical systems. Adaptability refers to the ability of a living system to adjust and thrive in response to changing environmental conditions, maintaining critical functions for survival and fitness. The brain, a complex dynamical system, plays a crucial role in animal adaptation^1–6.

The fundamental cognitive functions of decision-making (DM) and working memory (WM) are integral to animal adaptation. DM is the process of selecting among alternative courses of action based on available information and goals, enabling animals to balance risks and rewards, avoid threats, and obtain resources. WM allows for the temporary storage and manipulation of information, aiding in DM. From an evolutionary perspective, WM and DM are thought to have evolved as adaptations to improve the survival of organisms in their environments^7,8. Natural selection has shaped the neural circuitry that underlies these cognitive functions over time. Despite significant advances in the field^9–11, the mechanisms related to working memory and cognitive DM remain elusive.

DM is associated with the ramping activity of neurons in the parietal, prefrontal, or premotor cortex, reflecting the accumulation of evidence^3,12–15. The WM function is associated with stimulus-selective persistent activity, a sustained neural response specific to a particular stimulus that persists beyond its presentation, commonly observed in the same cortical areas^3,13,15–17. Neurophysiological studies have shown that neural circuits exhibiting persistent activity are likely capable of integrating stimuli and making categorical decisions^3,15. Moreover, DM and WM may share similar circuit mechanisms.

Traditional models of DM, such as the drift-diffusion model (DDM), conceptualize the DM process as a linear accumulation of evidence towards predefined decision boundaries, characterized by parameters like drift rate, boundary separation, and starting point^18,19. While effective, these models often abstract away the underlying neural dynamics, limiting their capacity to capture the complexities of neural interactions during DM and WM tasks. In contrast, recent theoretical studies show that the characteristic neural activity of DM and WM can be successfully described by an attractor network framework^3,15,20,21, which offers a nonlinear, network-based approach to modeling DM processes. This framework uses the concept of attractor states, which are stable patterns of neural activity and represent different decision outcomes. By modeling neural populations and their dynamic interactions, the attractor framework provides a more biologically plausible and comprehensive depiction of the neural mechanisms underpinning DM and WM.

In this framework, attractor networks contain two or more populations with self-excitation and are selective to different stimuli. These excitatory populations interact with each other through a common pool of inhibitory neurons^3,15,20,21. The neural circuits with such self-excitation and mutual inhibition architecture can generate the stimulus-selective persistent activity pattern for WM^16,17,22, and also ramping, categorical, and winner-take-all dynamics for perceptual DM^15,23. Inhibitory neurons are specialized cells that release neurotransmitters to decrease the activity of other neurons, serving as a critical component in the regulation of the dynamics of neural circuits. They are usually provided by a single pool of non-selective neurons lacking connection specificity^3,15,20,21. However, recent studies have found that both excitatory and inhibitory neurons in decision circuits are selective, suggesting the existence of functional subnetworks within inhibitory populations, similar to excitatory populations^5,24. These findings prompt inquiries into why, out of all potential circuit architectures capable of supporting DM, a distinct architecture characterized by these precise attributes and configurations, comprising functionally specialized subnetworks, emerges through learning. Furthermore, it raises questions about whether this architecture could improve the accuracy of DM and the resilience of working memory representations in the presence of distractions. If the selective-inhibition architecture cannot protect memory from interference during DM, are there additional mechanisms, apart from the static organization of the circuit, such as temporal gating inputs, that can further modulate cognitive function?

To address these questions, we investigated the mechanisms underlying the DM and WM functions using the attractor network framework. This approach can reproduce psychophysical and physiological results in perceptual DM and working memory tasks^13,25 and provide a natural candidate mechanism for persistent neural activity and winner-take-all competition leading to a categorical choice. However, noise is ubiquitous in the brain, introducing stochastic transitions between different perceptual or physiological states. To comprehensively understand this system, it is imperative to quantify the underlying attractor landscapes, which fundamentally determine the behavior of the system. Unlike previous studies that have primarily relied on qualitative illustrations of attractor landscapes^3,15,20, we emphasize the quantitative assessment of landscape features such as basin depths and barrier heights. By doing so, we can relate these quantified landscape characteristics to observable behavioral metrics, including decision time and transition rates in various cognitive tasks. This quantitative approach, inspired by thermodynamics and statistical mechanics, allows for more precise predictions and a deeper understanding of the neural mechanisms governing DM and WM.

Inspired by thermodynamics and statistical mechanics, physicists have introduced concepts such as “internal energy”, “free energy”, and “entropy” to describe the collective properties of complex systems like neural circuits and proteins^1,26,27. Hopfield explored the computational properties of neural circuits, for example, memory storage and retrieval, by constructing an “energy function” (a representation of the stability and dynamics of system states)^1,26. However, this approach only works for symmetrical neural circuits and fails for more realistic asymmetrical connections.

To meet this challenge, we have developed a potential landscape and flux framework to explore the dynamics of neural circuits involved in DM and WM^21,28,29. Within our framework, the “potential” is an abstract concept that represents an effective potential landscape. This mathematical tool aids in visualizing the system’s dynamics, offering insights into the stability of different activity patterns and the transitions between them. It is important to underscore that this effective potential is distinct from the actual metabolic energy consumption of neurons.

Moreover, neural circuits, akin to other biological systems, require energy to perform various vital functions^30,31. The biological energy usage is associated with metabolic costs incurred during the maintenance and transition of neuronal states, such as the ATP (adenosine triphosphate) consumed by ion pumps to re-establish ionic gradients after activity and the energy necessary for action potential generation. In our non-equilibrium potential and flux framework, we account for these biophysical energy considerations by examining the entropy production rate. This entropy production rate serves as a proxy for measuring the cost of biological energy that underlies biological functions^32–34. By integrating this measure, we gain novel insights into the role of energy supply in the context of these cognitive processes.

Here, we use a reduced version of the spiking neuronal network models comprised of integrate-and-fire types through a mean-field approach^3,15,35, which can reproduce most of the psychophysical and physiological results in delayed response DM tasks^13,15,25. By quantifying the underlying attractor landscapes, which refer to the landscapes of stable states into which the system settles and determine the behavioral responses of the system, we found that a circuit architecture with selective inhibition results in stronger resting states, improving DM accuracy. However, the weaker decision states in this circuit structure lead to less stable working memories against distracting stimuli. Our results show that presenting a ramping non-selective input during the initial phase of the delay interval in DM tasks can serve as a temporal modulation mechanism that enhances WM robustness against distractors. Maintaining robustness requires stabilizing the decision attractor landscapes, which our thermodynamic analysis shows has an associated energetic cost. Compared to alternatives like a constant non-selective input, a ramping input achieves robustness using less total thermodynamic cost over time. This temporal gating mechanism complements the selective inhibition architecture to dynamically emphasize either robustness or flexibility of working memory depending on specific demands. Overall, our approach provides a new paradigm for exploring the underlying mechanisms of various cognitive functions.

Results

Influence of the circuit architecture on DM and WM functions

Both the posterior parietal cortex (PPC) and prefrontal cortex (PFC) are key nodes that can exhibit characteristic neural activity of cognitive functions such as DM and WM^36–38. The perceptual decision-making behavior and working memory function in these cortical areas were well explored through the classic random-dot motion (RDM) discrimination tasks^39–42. In such experimental paradigms, monkeys are trained to judge the direction of motion of random dots, and their choice responses are indicated by a rapid saccadic eye movement. In a delayed response version of the task¹³, monkeys are required to withhold the response for a delay period, which means that its choice must be actively maintained in working memory. To clarify the neural foundation of DM and WM, models have shown that DM can be defined by ramping dynamics, where neural activity gradually increases over time as evidence accumulates, and a “winner-take-all” process in which the neural population representing the chosen choice suppresses the activity of other competing populations^3,20. Meanwhile, WM is linked to sustained activity within neural circuits that preserves stimulus information over a delay period^3,23. These circuits are structured so that groups of stimulus-selective excitatory neurons compete, regulated by feedback inhibition from a common pool of non-selective inhibitory neurons^{3,15,20,21,23} (Fig. 1a).

Fig. 1. Selective subnetwork of inhibitory neurons enhances the accuracy in decision-making.

a The diagram of the circuit model with non-selective inhibitory neurons. b The circuit architecture with a selective subnetwork of inhibitory neurons. The control parameter w can indicate the degree of selectivity of the inhibitory neurons. w and w₋ modulate the strength of connections between the excitatory and inhibitory populations. w increases the strength of connections between two sub-populations with the same subscript relative to cross-connections. w₋ decreases with increasing w, modeling the homeostatic depression of connections between clusters with the different subscripts (1,2). Larger w implies the inhibitory neurons more selective (see more details in the Methods section). c, d Neural activities during DM for different circuit architectures with non-selective and selective inhibitory neurons, where w = 1, 1.06, respectively. The blue curves represent the activities of the excitatory neural population 1 for correct trials, while the red curves for the error trials. The black trajectories mark the firing rate of population 1 for the trial, with median decision time. e, f The average decision times and correct rates in DM with varied architectures indicated by the parameter w. For the generality of the results, we explored the DM tasks with different difficulties (stimulus motion coherence c = 0.1, 0.2, 0.5, respectively). Each data point in Fig. 1e, f are averaged over 30,000 trials. The circuit architecture with a selective subnetwork of inhibitory neurons (larger w) results in longer decision time, with variability across trials represented by error bars (SEM). In addition, the accuracy in DM increases with w. The underlying mechanism can be reflected from the attractor landscapes showing that the selective-inhibition architecture leads to a stronger resting state, which extends the time of integration of evidences ((g–i) with w = 1.01, 1.04, 1.06, respectively and the nonzero-motion coherence c = 0.1).

However, recent experimental and modeling evidence suggests that specific functional circuits exist within inhibitory populations, analogous to the stimulus-selective circuits within excitatory populations^5,24. These inhibitory “subnetworks” can be thought of as distinct inhibitory circuits preferentially connected to different excitatory populations. Studies have found that during learning, the formation of selectivity in these inhibitory subnetworks takes place concurrently with the development of selectivity in excitatory populations (Fig. 1b)²⁴. To study why such circuit architecture is chosen from many different alternatives, we explored a biologically realistic attractor model to test how different circuit architectures influence performance in DM and further WM tasks.

In traditional drift-diffusion models (DDM), the decision-making process is modeled as a linear accumulation of sensory evidence towards predefined decision boundaries, with parameters such as drift rate, boundary separation, and starting point determining the decision dynamics^18,19. However, these models do not explicitly account for the underlying neural circuitry involved in DM and WM. In contrast, the attractor network framework conceptualizes DM as a nonlinear process driven by the dynamic interactions of neural populations. A DM task in this framework commences from a resting state, representing baseline neural activity before decision-making begins. Upon presentation of an external stimulus, the system’s trajectory through the high-dimensional neural state space is governed by network dynamics, including mutual excitation within selective neural pools and inhibition between them. Unlike the linear evidence accumulation in DDM, trajectories in the attractor framework can exhibit complex patterns, such as nonlinear pathways, which ultimately converge in one of the competing attractor states, stable neural activity patterns that correspond to specific decision outcomes^15,20.

This approach not only captures the stability and flexibility of neural representations during DM but also integrates working memory processes through persistent neural activity within attractor states. By quantifying landscape features such as basin depths and barrier heights between attractors, we aim to relate these features to quantifiable aspects of the model’s behavior observable in simulations. Here, we use a reduced version of the spiking neuronal network models comprised of integrate-and-fire types through a mean-field approach^3,15,35, which can reproduce most of the psychophysical and physiological results in delayed response DM tasks^13,15,25. In this model, the two excitatory neural populations (selective for rightward and leftward motion directions) receive external inputs from the visual motion stimulus. These inputs are linear functions of the motion coherence c (the percentage of random dots moving coherently, see details in the Methods section), which sets the bias of the input for one population over the other, depending on whether the motion stimulus is in the preferred or non-preferred direction of the cell. For a zero-coherence stimulus (c = 0), the two excitatory populations receive equal input. Under these conditions, making a decision becomes more challenging, as the decision-making process is predominantly influenced by random noise. For a high coherence c, there is a larger difference in the external input currents to the two excitatory populations. This implies that it is easier to make a decision.

In contrast to the assumption of non-selective inhibition in previous theoretical models (Fig. 1a), recent studies suggested that the subnetworks of selective inhibitory neurons support DM²⁴. As shown in Fig. 1b, we model this circuit architecture assuming selective inhibitory neurons by dividing the inhibitory neurons into two sub-populations (I₁ and I₂), each connected preferentially to one excitatory pool (E₁ or E₂). We use a control factor w to modulate the connection strengths between the excitatory and inhibitory sub-populations. The connection strengths within population 1 or 2 are larger by the factor w, while cross-population (between 1 and 2) connection strengths are smaller by a factor of w₋(details in the Methods section). In the presence of external stimulus in DM, the increasing factor w(>1) resulted in a gradual increase in the distance between the average neural activities in the two inhibitory sub-populations (Fig.S1 in the supplementary materials), which are similar to the two competing choice states represented by the neural activities in the two selective excitatory populations. Therefore, larger w implies that the sub-populations are more selective, while w = 1 corresponds to the completely non-selective case. In this model, a nonzero-motion coherence stimulus suggests that the excitatory population E₁ receives larger external input. A correct choice is reflected in the ramping neural activity of the population E₁ to a high-activity state while the population E₂ to a low-activity state. As shown in Fig. 1c, d, the circuit structure indicated by the factor w plays a crucial role in determining the time integration in perceptual decisions, here with the motion coherence c = 0.2. With the non-selective structure (w = 1), the neural activity of the population E₁ ramps relatively faster. With a selective structure (w > 1), the timescale of integration is relatively slower.

The easier tasks indicated by the larger motion coherence c always correspond to shorter decision times and higher accuracy. However, for a specific motion coherence condition, there is a tradeoff between the decision time and the accuracy, which is known as the speed-accuracy tradeoff (SAT) in DM^31,43. Unlike the drift-diffusion model, which utilizes explicit decision thresholds to trigger decisions, our attractor model embeds decision criteria implicitly within the system dynamics. In this framework, the decision criterion is represented by the system’s convergence to one of the stable attractors, with the basin of attraction determining the conditions required for convergence to a particular decision state. Therefore, the decision time here is defined as the time required for the system to transition from the resting state to a decision attractor in response to the external stimulus. Figure 1e depicts that the average decision time increases as the factor w is increased in different motion coherence conditions. Figure S2 shows the decision time distributions (kernel density estimates) for varying w. The longer integration times in decision processes are accompanied by higher accuracy (Fig. 1f). However, the overall performance depends not just on this speed-accuracy tradeoff, it also relies on the specific constraints of the decision task. For tasks with long required delays between trials, higher w allows better accuracy to accumulate given available time. But for tasks emphasizing speed on individual trials, lower w can enable faster decisions and superior performance.

To investigate the underlying mechanisms for the improved accuracy that results from the signal-selective architecture, we quantified the attractor landscapes that characterize the DM function with varying w (see details in the Methods section). As shown in Fig. 1g, there are two biased basins of attraction (decision attractors) corresponding to the two choices in the presence of the nonzero-coherence stimulus. The attractor corresponding to the correct choice is stronger, whereas the other is weaker. The system initiates at the resting state between the two decision attractors before the onset of the external stimuli. When an external stimulus is introduced, the resting state becomes unstable, compelling the system to evolve toward one of the two decision attractors. This process serves as the decision rule, where the system’s dynamics dictate the final decision outcome based on the shape of the potential landscape. Specifically, the decision criterion is achieved when the system’s state traverses the landscape barrier separating the resting state from the decision attractors, effectively reaching a stable attractor corresponding to a particular choice. This mechanism allows for quantitative predictions of decision-related measures. For instance, higher barriers result in longer decision times as the system requires more time to accumulate sufficient evidence to overcome the barrier.

As w is increased, inhibitory neurons become more selective, leading to a stronger resting state characterized by a local minimum in the potential landscape situated between the two competing decision attractors (Fig. 1g–i) with w = 1.01, 1.04, 1.06, respectively). Consequently, a higher w value results in a larger barrier that trajectories must overcome to reach a decision attractor. This landscape configuration directly predicts longer decision times in simulations, as increased evidence accumulation is necessary to traverse the higher barrier (Fig. 1e). Additionally, the slower evidence integration process afforded by higher barriers reduces decision errors induced by random noise, enhancing decision accuracy.

To provide a more comprehensive understanding of the parameter relationships, we show a three-dimensional graph (Fig. 2) illustrating how accuracy varies across different combinations of inhibitory selectivity w and motion coherence c. The surface plot reveals a clear threshold boundary where accuracy transitions to 100%, with the critical w value decreasing as motion coherence increases (the red curve). This relationship demonstrates how task difficulty and circuit architecture interact: for more challenging tasks (lower c), higher inhibitory selectivity is required to achieve perfect performance, while easier discriminations can reach maximal accuracy with less selective inhibition. At the threshold boundary, the system transitions from bistable dynamics supporting two competing attractors to a monostable regime where only the correct choice attractor remains. Figures S3, S4 show the phase-planes for different w with the motion coherence c = 0.4 and 0.5, respectively. When w exceeds this critical value for a given coherence level, the potential barrier maintaining the incorrect choice state disappears, making the system deterministically converge to the correct decision regardless of noise fluctuations. These findings highlight how neural circuits can adaptively tune selectivity parameters to optimize decision-making performance across varying task difficulties, suggesting a principled framework for understanding the evolution of inhibitory circuit architectures in cortical networks supporting perceptual decisions.

Fig. 2. 3D parameter relationships and the threshold curve.

Three-dimensional visualization of decision accuracy as a function of selective inhibition w and stimulus strength c. The surface highlights parameter regimes where selective inhibition w and stimulus strength c synergistically optimize accuracy. Critical w values (red curve) at which decision accuracy reaches 100% for varying coherence c. Higher c reduces the required w, demonstrating that stronger sensory evidence compensates for weaker selective inhibition.

Our results raise the possibility that the brain chooses the higher accuracy architecture among possible alternatives to support DM. This theoretical prediction is consistent with previous experimental findings that used machine learning techniques such as support vector machines (SVMs) and classifier analysis to decode choice-related signals from neural population data²⁴. The study showed that choice representations emerged simultaneously in excitatory and inhibitory populations during task learning, supporting our model result that selective inhibition circuitry improves decision accuracy by strengthening choice coding in both populations.

In some cases, for example, in the delayed response version of the visual motion discrimination task¹³, the choice must be actively maintained in the working memory. Robust WM requires shielding the interference from both internal noise and external distraction. To assess how the signal-selective structure influences the WM function, a distractor stimulus is applied during the delay period. The strength of this distractor stimulus is equal to the original visual motion stimulus, while the motion coherence condition is inverse (the excitatory population E₂ receives a larger external input). The distractor is introduced after the system has reached the decision attractor and is presented until a transition occurs. During this time, the network received an additional input intended to disrupt the sustained activity representing the initial stimulus. Figure 3a shows that it takes less time to make a different choice (switch to the other decision state) in the presence of the distractor stimulus as the factor w increases. The robustness of the decision attractor can be quantified through the barrier height between the decision attractors that is inferred from the landscape topography. As shown in Fig. 3b, the barriers separating the current decision attractor from the other one are reduced as the circuit architecture changes from the one with non-selective inhibition to the one with selective inhibition.

Fig. 3. The robustness of WM against distractors.

a The average transition time from the correct choice to the error one in the presence of distractors for different circuit architectures. The system has to go across a potential barrier to switch to another decision state. b The barrier heights (bh) inferred from the underlying landscape topography can measure the robustness of the decision state in WM. Both the average transition time and the corresponding barrier heights are reduced as the w is increased, which implies less robustness against distractors.

The transition rates between decision attractors in the presence of distractors are influenced by the barrier heights, with higher barriers reducing the likelihood of transitions and enhancing WM resilience against distractions. As w increases, the barriers separating the decision attractors are reduced (Fig. 3b). Concurrently, the intermediate “initial” state becomes more stable (Fig. 1h, i). This combination of effects leads to less robust memory states that can be more easily reverted to the stabilized initial state under distraction, enabling quicker flipping between decision representations (Fig. 3a). The increasingly stable intermediate attractor at higher w provides a gateway state that accelerates transitions between decision attractors when the system is perturbed from sustained choice representations. Our results suggest that the signal-selective structure is less robust against distractors. On the other hand, the weakened stability of the decision state due to the strengthened intermediate state implies that the initial errors are more likely to be corrected in the presence of the visual stimuli. This can further improve the accuracy of DM in addition to the increased accuracy of the initial choices. In summary, there is a tradeoff between the accuracy in DM and the robustness in WM.

Dynamical modulation of the tradeoff between decision-making and working memory functions

Our results suggest that the circuit architecture with selective inhibition enhances accuracy in decision-making while reducing the robustness of decisions held in working memory against distracting stimuli. This raises an interesting question: What is the underlying mechanism for protecting memory from interference in the circuit with a selective inhibition architecture? If the circuit architecture is specified, one possibility is that gating incoming stimuli might be achieved by upstream modulation signals to the circuit. To test this hypothesis, we applied an additional non-selective input (i.e., the same input to both selective neuronal populations) during the delay period of the DM tasks. We examined how this additional non-selective input influences the dynamical behavior and the underlying attractor landscape of the system.

In the delay period of the DM tasks, the presence of distracting stimuli can cause changes from the initial correct choice to the incorrect one. Figure 4a shows that in the presence of the distracting stimuli, the average transition time to the incorrect decision state from the initial correct one increases as the additional non-selective input (indicated by Δ>I₀) is increased. This implies that the increasing non-selective input can enhance the robustness of WM against distractors. When the distracting stimulus is presented for a limited duration (for example, 1s), the proportion of switches to the wrong choices from the initial correct one reduces with the increased non-selective input (Fig. 4b). The mechanism of the enhanced robustness against distractors due to the additional non-selective input can be uncovered from the changes in the underlying landscape topography (Fig. 4c). Both the two decision attractors become stronger with increased non-selective input. The larger non-selective input increases not only the separation between two decision attractors, but also the depth of the attractor basins and the barrier between them (Fig. 4d–f). It is less likely to go across the barrier and switch to the other decision state in the presence of a larger non-selective input, thereby resulting in the gating of the distracting inputs.

Fig. 4. An increasing non-selective input can enhance the robustness of WM against distractors.

a The average transition time from the correct choice to the error one in the presence of distractors increases with increased non-selective input to both selective excitatory neural populations. The ΔI₀ here represents the magnitude of the additional non-selective input. b The transition rate in a limited duration of the distracting stimulus (1 s) is reduced as the non-selective input is increased. The mechanism of improved robustness is reflected in the larger barrier height (bh) inferred from the underlying landscape topography (c). The detailed attractor landscapes with increased non-selective input are displayed in (d–f), where the non-selective input I₀ = 0, 0.004, 0.008 nA, respectively, with the specific nonzero-motion coherence c = 0.2 and w = 1.

In the delay period, the presence of distracting stimuli can cause transitions from the initial correct choice to the incorrect one. Figure 4a shows that, in the presence of distracting stimuli, the average transition time from the correct decision state to the incorrect one increases as the magnitude of the additional non-selective input increases. This implies that increasing the non-selective input enhances the robustness of WM against distractors. When the distracting stimulus is presented for a limited duration (e.g., 1 s), the probability of switching to the wrong choice from the initial correct one decreases with the increased non-selective input (Fig. 4b). The mechanism behind the enhanced robustness against distractors due to the additional non-selective input can be understood from changes in the underlying attractor landscape topology (Fig. 4c). Both the two decision attractors become stronger with increased non-selective input. The larger non-selective input increases not only the depth of the attractor basins for the decision states, but also the barriers between them (Fig. 4d–f). With larger barriers, it becomes less likely for the system to cross over to the other decision state in the presence of distractors, thereby effectively gating distracting inputs.

However, applying a large additional non-selective input directly is accompanied by increased thermodynamic cost. Neural circuits, as open systems with constant exchange of material, information, and energy with the environment, are intrinsically out of equilibrium. The thermodynamic cost for maintaining persistent neural activities can be measured by the entropy production rate in non-equilibrium systems^{29,32–34,44,45} (more details are provided in the Methods section). Figure 5a shows that this thermodynamic cost increases with the magnitude of the additional non-selective input. Therefore, while a larger non-selective input enhances WM stability against distractors, it does so at the expense of higher thermodynamic.

Fig. 5. The disadvantages of strong non-selective input in DM and WM.

Although the increased non-selective input can improve the robustness of WM against distractors. Larger thermodynamic costs are needed to support the enhanced robustness against distractors during the delay period (a). The ΔI₀ here represents the magnitude of the additional non-selective input. It leads to shorter decision time (b) while making less accurate choices (c), and the initial errors are less likely to be corrected (d). Presenting A ramping input during the delay period may serve as a cost-effective mechanism of temporal gating of distractors.

Moreover, we found that applying a large non-selective input throughout the entire task adversely affects the DM function. Specifically, when the additional non-selective input is applied from the onset of the visual stimuli during the evidence accumulation period of the DM tasks, it leads to faster decisions but with lower accuracy (Fig. 5b, c). The underlying landscapes that determine these behavioral responses are similar to the ones shown in Fig. 4f. The stronger decision attractors reduce the integration time of evidence, which can result in more errors. Additionally, the initial errors may be corrected during the delay period of the delay-response DM tasks^31,46. Figure 5d shows that the probability of correcting initial errors during the delay period decreases as the additional non-selective input increases. These results suggest that if the additional non-selective input is presented together with the target stimuli during the evidence accumulation period of the DM tasks, there could be more errors. This seems to be in contradiction with the mechanism underlying the circuit architecture with selective inhibition, in which the accuracy of DM should be emphasized. Thus, to preserve DM accuracy, the additional non-selective input should only be applied during the delay period to enhance the WM function against distractors without reducing DM accuracy.

To address the increased thermodynamic cost associated with a large constant non-selective input during the delay period, we propose that a gradual increase in the strength of the non-selective input during the delay period can achieve enhanced robustness more efficiently. Due to uncertainties in the exact temporal dynamics of such modulatory signals in biological systems, we introduce a specific time-varying form of the additional input (stepped increases over time) in our simulations. This increasing input allows for progressive strengthening of the decision attractors over time. Initially, the system maintains flexibility, which is beneficial for integrating any new relevant information and correcting potential initial errors. As the delay period progresses and robustness becomes more critical, the increasing non-selective input enhances the system’s resistance to distractors, effectively gating out interfering stimuli. By allocating energy resources in this time-dependent manner, the system can achieve stability with a lower overall thermodynamic cost. As demonstrated in Fig. S5, the ramping non-selective input reduces the transition rate (improving robustness) compared to the no-input condition while incurring lower entropy production than a large constant input. This mechanism dynamically prioritizes flexibility early in the delay period (allowing error correction) and robustness later (shielding against distractions). This approach complements the selective-inhibition circuit architecture and provides a dynamic mechanism to balance DM and WM functions according to cognitive task demands.

Discussion

One of the distinguishing features of biological systems, as opposed to conventional physical systems, is their ability to adapt to the environment. The fundamental cognitive functions of decision-making (DM) and working memory (WM) play crucial roles in processing and acting upon the information that animals perceive from their surroundings. Within the attractor network framework proposed by previous researchers, the distinct neural activities associated with DM and WM observed in the prefrontal and posterior parietal cortex can be replicated in the circuit architecture through strong recurrent excitation within selective populations and lateral inhibition via a single pool of inhibitory neurons^3,15,20. However, recent experimental and modeling evidence suggests the existence of specific functional subnetworks within inhibitory populations²⁴. These subnetworks challenge the previously proposed circuit mechanisms that assumed a single non-selective pool of inhibitory neurons, and the mechanisms underlying DM and WM functions based on such a circuit architecture remain unclear.

The attractor landscape provides a comprehensive and quantitative description of the dynamics of neural circuit systems. Different landscape topographies indicate that the number and relative weights of attractors (functional states) vary in the underlying landscapes, determining the categorical choice in decision-making and the robustness of working memory. By quantifying landscape features such as basin depths and barrier heights, we can directly relate these to observable behavioral metrics. For example, higher barrier heights necessitate longer decision times, as the system must accumulate more evidence to transition from the resting state to a decision attractor. This is in contrast to the drift-diffusion model, where decision thresholds are explicitly defined parameters. In the attractor framework, the decision criterion is implicitly embedded within the system’s dynamics, determined by the basin of attraction around each attractor. This allows for a more nuanced and biologically plausible representation of decision-making processes⁴⁷. Additionally, the transition rates between attractors, especially in the presence of distractors, are intrinsically governed by the potential barriers, enhancing our understanding of WM’s resilience against interference. These quantitative insights demonstrate the superiority of the attractor model in elucidating the underlying neural mechanisms of cognitive functions over more abstract models like the drift-diffusion model.

Although the concept of attractor landscapes has been extensively introduced to describe cognitive functions, they are often presented as illustrations^3,15,20 or quantified only in limited, specific circuits²⁶. Drawing inspiration from thermodynamics and statistical mechanics for physical systems, we have developed a non-equilibrium potential landscape and flux framework for general neural circuits^21,28,29. By applying this approach to a biophysically based model that can perform DM and WM computations, we have explored the circuit mechanisms of DM and WM by focusing on the influence of varied circuit structures on the system’s behavioral responses and underlying attractor landscapes.

By relating quantifiable natures of the potential landscape to computational features emergent in the simulations, our approach provides a comprehensive framework for elucidating mechanisms that govern the model’s function. We found that the circuit architecture with selective inhibition supports three stable states, including one resting state and two decision states, in the presence of visual stimuli. In contrast, the circuit architecture with non-selective inhibition only supports two decision states, resulting in a shorter time for evidence integration. Consequently, the decision time is longer with higher accuracy in the circuit architecture with selective inhibition, which is consistent with previous results obtained using a machine learning method²⁴.

We found that the circuit architecture with selective inhibition supports decision-making through competition between two choice attractors. Similar to models such as the leaky competing accumulator⁴⁸, this competition is implemented via winner-take-all dynamics from the selective inhibition. Specifically, selective versus non-selective inhibition results in a stronger intermediate resting state in the underlying potential landscape. This increases the evidence required to switch from rest to a decision, lengthening decision times while filtering out noise to improve accuracy for difficult choices. In both cases, inhibition between competing choices balances the speed-accuracy tradeoff through modulating evidence accumulation over time.

Protecting the decision held in WM from interference is crucial for cognitive tasks, such as a delayed response version of the decision-making task. However, whether the stability of WM can also benefit from such circuit architecture is still unknown. Our findings suggest that memories are less robust against distractors due to reduced barriers between decision attractors on the underlying landscapes. Thus, there is a tradeoff between the DM and WM functions based on a specific circuit architecture. Our prediction suggests that additional dynamic modulation mechanisms beyond the specific circuit architecture may be necessary for balancing DM and WM functions according to cognitive task demands.

Based on the selective-inhibition circuit architecture, which emphasizes DM accuracy, we found that applying an additional non-selective input during the delay period of the task enhances working memory robustness against distractors in a cost-effective manner. A larger non-selective input induces stronger decision attractors and higher barriers between them, making it less likely for distractors to change the initial choice. Our potential landscape and flux approach provides a valuable framework for understanding the dynamics of neural circuits.

In reality, the functioning of neurons is powered by metabolic energy, with ion pumps consuming ATP to maintain ionic gradients and action potentials representing a significant energy expenditure. We employ the concept of the entropy production rate to provide a realistic measure of the dissipation costs associated with neural activity. This concept is grounded in the principles of non-equilibrium thermodynamics and offers a direct connection to the thermodynamic expenditures required to sustain the neural network’s functions^32–34. We observed that maintaining higher barrier heights for enhanced robustness incurs a significant thermodynamic cost, as evidenced by the increased entropy production rates required to sustain these landscape configurations. This tradeoff underscores the balance between thermodynamic cost efficiency and cognitive performance, suggesting that the brain must optimize landscape characteristics to achieve the desired behavioral outcomes within metabolic constraints.

In terms of DM function, additional non-selective inputs can lead to shorter decision time but lower accuracy when presented together with target stimuli during the evidence accumulation period. Thus, the additional non-selective input should only be presented during the delay period against distractors. Valuable information may be transmitted to the DM circuit during the early delay period, which can help correct errors that may occur in the initial choice, and should not be gated. Combining these findings, the most efficient way to achieve higher accuracy in DM while guaranteeing a more robust choice with less cost is through the circuit architecture with selective inhibition and applying an additional non-selective input during the delay period in a manner that gradually increases in strength. These results are consistent with recent experimental work suggesting that a ramping input, rather than an input with constant strength, renders the system insensitive to distracting stimuli over time⁴⁹.

Sometimes, in order to adapt to environmental conditions or demands from behavioral tasks, the working memory system should be highly flexible rather than robust to incoming stimuli^22,50. Previous research has investigated the biophysical mechanisms responsible for this flexibility, including the ability to erase memory (return to the resting state) using a negative input⁵¹. A recent study suggests that an intermediate state with high activity for both selective neural populations, resulting from increased recurrent excitation within these neural pools, can enhance flexibility to new stimuli²¹. In this study, we found that transitions between different memory states can be promoted through circuit architecture with selective inhibition, where the resting state serves as the intermediate state for switching. This intermediate state increases the accuracy of initial choices and may help correct initial errors in the presence of visual stimuli from the perspective of the DM function. From the perspective of the WM ability, such an intermediate state makes the circuit more flexible to the most recent incoming stimulus, making it easier to load new inputs and discard old memories. This implies a new network mechanism, indicating that enhanced flexibility to new stimuli in WM does not require an additional negative input to reset the system, nor does it require the intermediate state resulting from increased recurrent excitation. Rather, enhanced flexibility can arise from circuit architecture with selective inhibition, resulting in a strengthened resting state as the intermediate state for transitions to another memory state. Future experiments monitoring neural activity in subjects trained to remember the latest stimulus will reveal whether enhanced flexibility originates from the selective inhibition circuit.

A recent modeling study has also highlighted the crucial role of inhibitory neurons in DM, with their specificity in connections that mediate the speed-accuracy trade-off⁵². Our current work examines not only how the selective inhibition circuit architecture affects DM function but also the tradeoff between DM and WM functions based on this specific circuit architecture. Additionally, we investigate potential dynamic modulation mechanisms beyond the specific circuit architecture that can balance DM and WM functions according to cognitive task demands. Importantly, our potential landscape and flux approach provides explicit elucidation of the biophysical mechanisms underlying specific cognitive functions, in contrast to previous studies relying only on statistics of individual trajectories^23,24,53. Furthermore, our approach quantifies the nature of dissipative consumption underlying cognitive functions.

Recent advances have expanded landscape quantification approaches^21,28,31 to encompass both spiking neural networks and large-scale working memory models^54,55. These works demonstrate the utility of landscape frameworks in analyzing complex anatomical architectures and distributed neural systems. However, our simplified architecture intentionally isolates core circuit mechanisms, such as selective inhibition, to systematically examine how circuit parameters shape attractor landscapes, thermodynamic costs, and cognitive tradeoffs. This targeted approach enables us to establish direct causal relationships between inhibitory subnetworks and cognitive functions, which can be challenging to disentangle in models with extensive connectivity patterns or region-specific variations. Our work provides mechanistic insights into a foundational cortical processing module, complementing broader network analyses by revealing circuit-level principles that may underlie distributed cognitive computations⁵⁶. Future research can integrate these findings with detailed connectomic data to investigate how selective inhibition architectures interact with long-range connectivity patterns to optimize cognitive performance across distributed neural systems.

Overall, our results indicate that fundamental evolutionary drivers for neural circuit architectures supporting DM and WM have likely emphasized the optimization of DM accuracy and the enhancement of WM flexibility. This evolutionary shaping allows for an effective response to environmental changes, with the circuit structure featuring selective inhibition fine-tuned to balance behavior and optimize DM based on the latest information. When the behavioral task demands prioritize memory robustness against distractors, applying an additional non-selective input during the delay period, in a manner that gradually increases in strength, can enhance the robustness of WM against distractors with a relatively lower thermodynamic cost compared to maintaining a constant high input throughout the delay period. This temporal gating mechanism complements specific functional subnetworks in the circuit architecture to achieve an optimal balance between the DM and WM functions.

Our study demonstrates that the non-equilibrium landscape and flux framework offers a comprehensive and quantitative approach to understanding the neural mechanisms of DM and WM. The ability to quantify landscape features provides a powerful tool for linking neural circuitry parameters to cognitive function metrics. For instance, adjusting the selectivity parameter offers a direct pathway to modulate decision accuracy and WM robustness, offering targets for future experimental investigations. By explicitly linking landscape features to decision time and transition rates, we provide a robust foundation for future explorations into the dynamic modulation of cognitive functions within biologically plausible neural architectures.

Methods

Non-equilibrium landscape and flux framework for general neural circuits

The metaphor of an attractor landscape is employed to conceptualize how neural activity patterns correspond to various cognitive states, with each state represented as a basin or valley within this landscape. For example, this attractor landscape metaphor is widely used to describe cognitive functions such as associative memory retrieval, classification^1,26,57. However, this concept can only be mathematically delineated and computed, in terms of stability and transition rates, in models with specific symmetry assumptions, such as the symmetrical neural circuit in the original Hopfield model^1,26. To address the issues of global stability of states associated with cognitive functions, we developed a non-equilibrium potential landscape and flux theory for a wide array of neural network models beyond those with specific symmetries, thereby encompassing more biologically realistic architectures²⁸.

For realistic complex systems comprising numerous interacting components, tracing microscopic trajectories of every component via solving corresponding dynamical equations can be a daunting task. Nonetheless, macroscopic emergent behaviors of the system can be approximately described by the dynamical evolution of several macroscopic observables with deterministic driving forces and random forces that represent effects of microscopic degrees of freedom on macroscopic observables. One well-known example is the stochastic Langevin particle dynamics. We consider a system described by a set of Langevin equations: $\frac{d\mathbf{x}}{dt}=\mathbf{F}\left(\mathbf{x}\right)+\mathbf{\zeta }$ in which the vector, x, represents the state variables of the system. The corresponding driving force is represented by the vector, F(x), while ζ represents Gaussian white noise, which has an ensemble average (denoted by < ζ > = 0) of zero, signifying that the mean of the noise over many realizations is zero. The autocorrelations $<\mathbf{\zeta }\left(\mathbf{x},t\right)\mathbf{\zeta }\left(\mathbf{x},t^{\prime }\right)>=2D\mathbf{D}\left(\mathbf{x}\right)\delta \left(t-t^{\prime }\right)$. Here, D is a scale factor representing the magnitude of the fluctuations, and D(x) is the diffusion tensor or matrix. The state variables x typically evolve according to nonlinear dynamical laws due to the complex interactions in the system. These laws can lead to diverse behaviors, and, when coupled with stochastic influences from random forces (noise), result in trajectories that are inherently unpredictable.

These stochastic differential equations describe the neural firing trajectories in time, the corresponding probabilistic evolution is given by the Fokker–Planck equation for the probability density^28,58,59:

$$\begin{array}{cc}\hfill \partial P\left(\mathbf{x},t\right)/\partial t=& \hfill \nabla \cdot \left(\nabla \cdot \left(D\mathbf{D}\left(\mathbf{x}\right)P\left(\mathbf{x},t\right)\right)\right)\hfill \\ \hfill & \hfill -\nabla \cdot \left(\mathbf{F}\left(\mathbf{x}\right)P\left(\mathbf{x},t\right)\right)\hfill \end{array}$$ 1

The corresponding Fokker–Planck equation can be written as:

$$\partial P\left(\mathbf{x},t\right)/\partial t=-\nabla \cdot \mathbf{J},$$ 2

where the probability flux is defined as:

$$\mathbf{J}=\left(\mathbf{F}\left(\mathbf{x}\right)*P\left(\mathbf{x},t\right)\right)-\nabla \cdot \left(D\mathbf{D}P\left(\mathbf{x},t\right)\right)$$ 3

The change in the local probability P(x, t) over time is determined by the net probability flux J flowing in or out of that region, ensuring local conservation of probability. The divergence of the flux is zero at long times when a steady state emerges:

$$\nabla \cdot {\mathbf{J}}_{SS}=-\partial P{\left(\mathbf{x}\right)}_{SS}/\partial t=0,$$ 4

J_SS represents the steady-state flux. However, it is not necessary for the flux itself to be zero under this divergence-free condition. A nonzero flux signifies a net directional flow of probabilities that disrupts the detailed balance—a state where every microscopic transition between states is exactly counterbalanced by its reverse—highlighting that the system is operating out of thermodynamic equilibrium. Additionally, the divergence-free flux suggests that there are no sources or sinks for the probability to go to or come from, and hence, locally, the flux must be rotational.

When a system reaches steady-state, the corresponding steady-state probability distributions can be solved through the steady-state equation:

$$\nabla \cdot \left[\mathbf{F}\left(\mathbf{x}\right)P_{SS}\left(\mathbf{x}\right)-\nabla \cdot \left(D\mathbf{D}\left(\mathbf{x}\right)P_{SS}\left(\mathbf{x}\right)\right)\right]=0$$ 5

Further, the non-equilibrium potential function $U=-ln{P}_{SS}\left(\mathbf{x}\right)$ can be defined in analogy to the Boltzmann law in equilibrium statistical mechanics. The driving force can be decomposed as^{28,29,31,58,59}:

$$\begin{array}{cc}\hfill \mathbf{F}& \hfill ={\mathbf{J}}_{SS}/P_{SS}+D\mathbf{D}\cdot \nabla \left(P_{SS}\right)/P_{SS}+\nabla \cdot D\mathbf{D}\hfill \\ \hfill & \hfill ={\mathbf{J}}_{SS}/P_{SS}-D\mathbf{D}\cdot \nabla U+\nabla \cdot D\mathbf{D}.\hfill \end{array}$$ 6

In contrast to equilibrium systems, where the dynamics are governed solely by the gradient of the underlying energy function, resulting in a zero net flux, the dynamics of non-equilibrium systems are influenced by both the potential gradient and the curl flux (J_SS). The non-equilibrium potential U serves as a useful tool for characterizing the global behavior of such systems.

In a steady state, non-equilibrium thermodynamic systems are associated with continuous entropy production^28,29,31. While the entropy of a non-equilibrium system in steady state remains constant over time, there is a flow of entropy to the surroundings equal to the entropy generated spontaneously within the system. To calculate the entropy production, the entropy associated with the time-dependent probability distribution can be focused on. Using the Fokker–Planck equation and the definition of probability flux J(x), the time derivative of the system entropy dS/dt can be expressed as a sum of two terms:

$$\frac{d}{dt}S\left(t\right)=\int d\mathbf{x}\left(\mathbf{J}\cdot {\left(D\mathbf{D}\right)}^{-1}\cdot \mathbf{J}\right)/P-\int d\mathbf{x}\left(\mathbf{J}\cdot {\left(D\mathbf{D}\right)}^{-1}\cdot \left(\mathbf{F}-D\nabla \mathbf{D}\right)\right).$$ 7

Notice that the first term on the right-hand side of the equation is always larger or equal to zero due to the positive definite diffusion matrix D. Therefore, it can be identified as the total entropy production rate, which is the global thermodynamic dissipation or cost directly linked to the non-equilibrium flux from the dynamics²⁹:

$$\mathrm{EPR}={\dot{S}}_{\mathrm{tot}}=\int d\mathbf{x}\left(\mathbf{J}\cdot {\left(D\mathbf{D}\right)}^{-1}\cdot \mathbf{J}\right)/P.$$ 8

where EPR stands for entropy production rate. The second term on the right-hand side of the equation 6 can then be regarded as the entropy flux from the system to the environment:

$${\dot{S}}_{\mathrm{env}}=\int d\mathbf{x}\left(\mathbf{J}\cdot {\left(D\mathbf{D}\right)}^{-1}\cdot \left(\mathbf{F}-D\nabla \mathbf{D}\right)\right).$$ 9

The total entropy change is equal to the sum of the entropy change of the system and that of the environment, leading to the emergence of the generalized non-equilibrium first law of thermodynamics:

$${\dot{S}}_{\mathrm{tot}}=\mathrm{EPR}=\dot{S}+{\dot{S}}_{\mathrm{env}}.$$ 10

The entropy change of the non-equilibrium system (${\stackrel{°}{S}}_{sys}=\stackrel{°}{S}$) can either be increased or decreased due to the entropy flow to the environments, while the total entropy change of the system and the environments, represented by the entropy production rate, is always non-negative. This leads to the generalized non-equilibrium thermodynamic second law:

$${\dot{S}}_{\mathrm{tot}}\ge 0.$$ 11

Quantification of neural circuit energy expenditure

Biological systems, such as neural circuits, consume energy to perform different vital functions^30,31. While increased firing rates generally lead to increased energy demand⁶⁰, relying solely on absolute firing rate to quantify neural circuit energy expenditure fails to capture substantial portions of the brain’s overall energy budget. As Marcus E. Raichle described with the concept of “dark energy,” a considerable fraction is devoted to baseline cellular processes independent of external stimuli, with task-related increments potentially accounting for only 0.5–1.0% of total consumption⁶¹. Additionally, nearly one-third finances “housekeeping” functions such as protein and lipid synthesis, proton leak across the mitochondrial membrane, and cytoskeletal rearrangements. Firing rates do not reflect these continuous, non-signaling processes comprising 25–50% of brain energy expenditure⁶². Hence, firing rate alone may not adequately measure the spectrum of neural circuit energy costs.

In our study, we employ the concept of the entropy production rate (EPR) to quantitatively describe the energy expenditure of biological systems—specifically neural circuits—by mapping their continuous energy dissipation. Our study is grounded in a robust theoretical framework, supported by a series of both theoretical and empirical studies^{29,33,34,44,63,64}, that positions EPR as a pivotal thermodynamic quantity capable of quantifying the continuous energy dissipation characteristic of biological systems.

To operationalize this concept, we consider the biological system as an open system that is maintained far from equilibrium by the constant input and dissipation of energy, primarily through the hydrolysis of ATP and GTP molecules. The total thermodynamic cost is contributed both from the free energy input or known as the housekeeping heat Q_HK⁶⁵ to maintain the non-equilibrium steady state and the generalized free energy relaxation $-\frac{d\mathcal{F}}{dt}$: $T\cdot \mathrm{EPR}=Q_{\mathrm{HK}}-\frac{d\mathcal{F}}{dt}$. The total thermodynamic dissipation or cost is quantified as the product of EPR and temperature, encompassing both steady-state maintenance and free energy changes^29,44,63.

The hydrolysis of ATP, a main source of entropy production, provides the free energy to drive endergonic biochemical reactions, maintaining non-equilibrium conditions in living cells. The ATP/ADP ratio, indicative of these conditions, results in a chemical potential difference $\mathrm{\Delta }G~ln\left(\mathrm{ATP}/\mathrm{ADP}\right)$, analogous to the voltage in electric circuits, which generates a non-equilibrium flux, as the energy pump of the system³². As we have shown in Eq.8: EPR = ∫ dx(J ⋅ (DD)⁻¹ ⋅ J)/P, the EPR is directly linked to the non-equilibrium flux. The previous work shows that the non-equilibrium flux J is directly linked to the chemical reaction flux³². The rate of ATP/GTP hydrolysis and the consumption of other metabolic fuels are thus directly related to the EPR, which quantifies the minimal rate of heat dissipation required to counteract fluctuation-driven disorder. Greater EPR denotes higher expenditure and metabolic demand to sustain non-equilibrium living system dynamics. Notably, this non-equilibrium entropy measure stems from stochastic thermodynamics considerations of dissipative work rather than information-theoretic entropy reflecting the state uncertainty and informational capacity more commonly applied in computational neuroscience contexts⁶⁶.

Crucially, the relevance of EPR as a measure of energy cost is empirically supported by studies on the Escherichia coli chemosensory system³³. In this pivotal study, researchers used the entropy production rate to quantify energy consumption, revealing a fundamental energy-speed-accuracy relationship. They showed that the non-equilibrium state of the system’s negative feedback mechanism necessitates energy dissipation for accurate sensory adaptation. The experimental validation of theoretical predictions in E. coli chemotaxis confirms the utility of EPR in quantifying energy costs in biological adaptation mechanisms.

Our application of EPR to neural circuits is informed by these findings, providing a robust framework for assessing continuous and non-equilibrium energy demands. This methodological advance allows us to quantify energy dissipation beyond transient neuronal firing, offering a more holistic view of the brain’s energy utilization.

The neural circuit model

In this study, the decision to utilize an attractor network framework over the drift-diffusion model stems from the primary objective of elucidating the underlying neural mechanisms of cognitive functions such as decision-making and working memory. While the DDM offers a robust mathematical framework for modeling behavioral data like reaction times and choice probabilities, it remains abstract in terms of neural implementation. The attractor model, on the other hand, provides a biologically plausible representation of neural dynamics by incorporating features like selective inhibition and mutual excitation among neural populations. By choosing the attractor model, our study aims to bridge the gap between cognitive theories and neural substrates, offering a more comprehensive understanding of how the brain processes and maintains information during decision-making and memory tasks.

This study focuses on the neural circuit, where the collective behavior of the network is of greater significance than the individual neurons. Therefore, the model developed for this study utilizes average firing rates instead of individual spikes, which is well-suited for representing populations or nuclei within a network. The corresponding mean-field approach, which reduces the spiking model to the firing rate model, is based on the seminal work of Wilson and Cowan⁶⁷, which was later refined by Dayan and Abbot⁶⁸ to address the computational and interpretational challenges posed by other models. The state variables that describe the macroscopic emergent behaviors of the neural circuit in the present model are the firing rates of different nuclei.

Here we explored a biophysics-based model that is able to perform WM and DM computations^3,15,53,67. The model contains two selective, excitatory populations, labeled E₁ and E₂. These two populations have self-excitations from the strong recurrent excitatory connections within each excitatory population, and the overall effective connectivity between the two excitatory populations is inhibitory through the inhibitory population³. In contrast to the original models showing that the excitatory populations receive inhibition from a common pool of inhibitory interneurons^3,15, recent studies suggest the existence of subnetworks of selective inhibitory neurons in the decision-making circuit²⁴. We divide the inhibitory neurons into two sub-populations, labeled I₁ and I₂, as shown in Fig. 1b. To manipulate the degree of selectivity of inhibitory neurons, we set the connection strengths between two sub-populations with the same subscript to be stronger by multiplying a control factor w, while the connection strengths between sub-populations across different populations are made smaller by multiplying the factor $w_{-}=1-\frac{f\left(w-1\right)}{1-f}$, where f = 0.45. The factor w₋ is chosen based on previous work that describes how the synaptic weights between neurons belonging to different clusters were decreased⁶⁹. The value of w determines the selectivity of the inhibitory neurons: w = 1 corresponds to non-selective inhibition, while a larger w implies more selective inhibition.

Previous modeling studies suggest that the decision-making circuit, which consists of thousands of spiking neurons, can be reduced into a two-variable model through several approximations, such as the linearization of the input-output relation of the inhibitory interneurons^3,15. Based on the fact that the synaptic gating variable of NMDA receptors has a much longer decay time constant than other timescales in the circuit, the dynamics of the system can be described by the dynamical equations of the average NMDA synaptic gating variable S_i:

$$\frac{d{S}_i}{dt}=-\frac{S_i}{{\tau }_S}+\left(1-S_i\right)\gamma f\left(I_{i,\mathrm{tot}}\right),i=1,2,$$ 12

where firing rate r_i of neural population i is a function of total input current I_i,tot that can be written as^15,53:

$$r_i=f\left(I_{i,\mathrm{tot}}\right)=\frac{a{I}_{i,\mathrm{tot}}-b}{1-exp\left[-d\left(a{I}_{i,\mathrm{tot}}-b\right)\right]},i=1,2$$ 13

The corresponding parameters are a = 270(VnC)⁻¹, b = 108 Hz, d = 0.154 s, γ = 0.641, and τ_S = 100 ms, the same as those shown in the previous work¹⁵. The total synaptic input currents of the two neural populations dominated by the NMDA receptors are:

$$\begin{array}{cc}\hfill I_{1,\mathrm{tot}}=& \hfill w_0*J_{\mathrm{EE}}*S_1+w_{-}*J_{\mathrm{EE}}*S_2-w*J_{\mathrm{IE}}*I_{I1}\hfill \\ \hfill & \hfill -w_{-}*J_{\mathrm{IE}}*I_{I2}+I_{\mathrm{BG}}+I_{1,\mathrm{ext}}+\zeta \hfill \\ \hfill I_{2,\mathrm{tot}}=& \hfill w_0*J_{\mathrm{EE}}*S_2+w_{-}*J_{\mathrm{EE}}*S_1-w*J_{\mathrm{IE}}*I_{I2}\hfill \\ \hfill & \hfill -w_{-}*J_{\mathrm{IE}}*I_{I1}+I_{\mathrm{BG}}+I_{2,\mathrm{ext}}+\zeta \hfill \\ \hfill \end{array}$$ 14

here J_EE is the effective coupling constant between excitatory sub-population and J_IE is the coupling constant from the inhibitory sub-population to the excitatory sub-population for the non-selective case(w = 1). The factor w₀ = 1.7 indicates the neurons belonging to the same excitatory sub-population with potentiated synaptic weight. ζ is the Gaussian noise term. With the linear approximation of the input-output transfer function of the inhibitory cell, the inhibitory input from an inhibitory sub-population to an excitatory sub-population depends on the total synaptic input current(I_I1 or I_I2) that this inhibitory sub-population receives, where

$$\begin{array}{cc}\hfill & \hfill I_{i1}=w*J_{\mathrm{EI}}*S_1+w_{-}*J_{\mathrm{EI}}*S_2-w_{-}*J_{\mathrm{II}}*I_{I2}\hfill \\ \hfill & \hfill I_{i2}=w*J_{\mathrm{EI}}*S_2+w_{-}*J_{\mathrm{EI}}*S_1-w_{-}*J_{\mathrm{II}}*I_{I1}\hfill \\ \hfill \end{array}$$ 15

Substitute Equation (15) into Equation (14), and we can obtain the total synaptic input currents I_1,tot and I_2,tot that depend only on the average NMDA synaptic gating variables S₁ and S₂. It can be easily proven that the corresponding parameters in the input-output transfer function of the inhibitory cell can be absorbed into the coupling constants J_IE, J_EI, J_II, and the background input I_BG. Here, the particular values of the parameters are set as J_EE = 0.48 nA, J_II = 5 nA, J_EI = 0.52 nA, J_IE = 1 nA, and I_BG = 0.31 nA based on the refs. ^3,15,24.

To characterize the DM function in this circuit model, the external stimulus input to the selective neural population is introduced as I_i,ext = J_A,ext ⋅ μ ⋅ (1 ± c), where i = 1, 2, J_A,ext = 5.2 × 10⁻⁴ nA/Hz, μ = 30 Hz is the external average synaptic coupling with the AMPA receptors. The + or − sign refers to whether the stimulus is the preferred or non-preferred one of the corresponding selective neural population. As introduced in the main text, the motion coherence c sets the bias of the input for one population over the other. For a zero-coherence stimulus (c = 0), the two excitatory populations receive equal input.

To quantitatively analyze the neural network dynamics underlying DM and WM functions, we apply the non-equilibrium potential landscape and flux framework to this biophysically based neural network model. The stochastic differential equations above can be translated into a Fokker–Planck equation describing the evolution of the probability density function P(I_1,tot, I_2,tot, t). Then we can calculate the probability distributions at steady state: P_SS and the corresponding potential function $U=-ln{P}_{SS}$. Furthermore, we can obtain the non-equilibrium flux J_SS and the corresponding entropy production rate EPR = ∫ dI_tot(J ⋅ (DD)⁻¹ ⋅ J)/P.

In our attractor network model, tasks such as perceptual decision-making and working memory are simulated through the dynamics of neural populations. Decision rules are inherently defined by the system’s transition from a resting state to one of the stable attractor states in response to external stimuli. Specifically, when an external stimulus is introduced, the resting state becomes unstable, compelling the system to evolve towards one of the two decision attractors. This transition constitutes the decision-making process. Unlike the drift-diffusion model, which employs explicit decision boundaries or thresholds to determine when enough evidence has been accumulated to make a choice, our attractor model uses the potential landscape to implicitly govern decision criteria. The basin of attraction around each attractor determines the conditions under which the system will converge to that attractor, effectively setting the decision criterion without the need for predefined thresholds. Specifically, the decision process continues until the system reaches one of the decision states. This is detected using detection circles placed around the two decision minima. As soon as the evolving activities reach one of these circles, a choice is made, and the decision part of the trial is completed. The choice pertains to the associated decision minimum. The radius of this circle is one-tenth of the distance between two minima in the potential surface, which balances sensitivity to fluctuations in neural activity while preventing premature decisions.

The WM function is characterized by stimulus-selective persistent activities and their robustness against distractors. The distractor-related current is stimulated as an input with the same strength as the original external stimulus, while the coherence condition is inverse (the excitatory population that initially receives a larger external input now receives a smaller input). To investigate the robustness of WM against distractors, this distractor-related current was applied throughout the delay period until transitions occurred in our simulations. The delay period represents the phase where the working memory must maintain the choice in the absence of the initial stimulus.

We also examined the effect of an additional non-selective input on WM and DM functions. Due to uncertainties in the temporal dynamics of these modulatory signals in biological systems, we did not specify a time-varying form for the input in our simulations. For the WM function, a constant non-selective input was applied to both selective neuronal populations during the delay period until transitions from the original decision attractor occurred. By varying the input’s magnitude, we assessed its influence on the stability of working memory representations against interference. In the case of DM, the non-selective input was applied alongside the external stimulus throughout the decision-making process until a decision was reached.

The dynamical trajectories of the system are calculated using the Runge–Kutta method with an integration time step of 0.02 ms. In each trial, the random fluctuation is introduced by a noise term implemented as uncorrelated standard Gaussian noise with zero mean and a variance equal to 0.024 nA. For the computations of the average decision time/transition time and correct rate, each data point is obtained from more than 30,000 trials.

Author contributions

H.Y. and J.W. designed research; H.Y. and J.W. performed research; H.Y. contributed new reagents/analytic tools; H.Y. and J.W. analyzed data; and H.Y. and J.W. wrote the paper.

Data availability

The data that support the findings of this study are openly available on the open science framework at https://osf.io/hw5rd/ (10.17605/OSF.IO/HW5RD).

Code availability

The code that support the findings of this study are openly available on the open science framework at https://osf.io/hw5rd/ (10.17605/OSF.IO/HW5RD).

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41540-025-00520-2.