Alpha-hemolysin is an exotoxin secreted by the bacterium Staphylococcus aureus. In its active form, the protein is built from 293 water-soluble amino acids that assemble into a heptameric pore, a seven-part barrel that punches through cell membranes. That structure, first resolved in crystallographic detail in the 1990s, has made alpha-hemolysin the workhorse of stochastic sensing: the first and still most widely used biological nanopore for detecting individual molecules. When a single pore is embedded in an artificial membrane bathed in electrolyte and a voltage is applied, a steady ionic current flows through the opening. When an analyte molecule wanders into the pore, it partially obstructs that current, producing a characteristic downward blip. Each blip corresponds to one molecule, and the pattern of blips over time carries information about the analyte’s identity and concentration.

In the new experiments, the team used monodisperse polyethylene glycol of molecular weight 1294, abbreviated PEG 1294, as the analyte. The experimental platform was a solvent-free planar lipid bilayer assembled by the classic Montal and Mueller technique: two monomolecular lipid films were apposed across a hole roughly 100 micrometers in diameter in a Teflon partition, forming a barrier that separated two identical compartments of a Teflon chamber. The synthetic lipid 1,2-diphytanoyl-sn-glycero-3-phosphocholine was used at 2 percent by weight in hexane, and the hole was pretreated with a dilute hexadecane solution to promote stable bilayer formation. Both compartments held a concentrated electrolyte of 4 molar potassium chloride buffered with 5 millimolar Tris-citric acid at pH 7.5. Alpha-hemolysin was added to one side, the cis side, at a concentration tuned so that exactly one pore inserted into the membrane.

The researchers then added PEG 1294 to the opposite, trans side at two concentrations, 400 and 1000 micromolar, and applied voltages ranging from +20 to +100 millivolts in 20 millivolt steps. Ionic currents were recorded with an Axopatch 200B amplifier in voltage clamp mode, using silver/silver chloride electrodes connected through agar salt bridges. The signals were low-pass filtered at 15 kilohertz and digitized at a sampling rate of 250 kilohertz, all at a controlled temperature of 25 degrees Celsius. Under these conditions, the current trace showed the textbook signature of stochastic sensing: stepwise transitions between a fully open level and a partially blocked level, each step corresponding to the entry or exit of a single PEG molecule from the pore lumen.

The novelty of the study lay not in the electrophysiology but in the analysis. Rather than focusing on conventional metrics such as event duration or blocking depth, the team turned to Lempel-Ziv complexity, a measure introduced in 1976 by Abraham Lempel and Jacob Ziv to quantify the randomness of finite sequences. The method works by converting a time series into a binary string: each data point is compared with the average of the entire series, receiving a 1 if it exceeds the average and a 0 if it falls below. The resulting string of ones and zeros is then scanned from left to right, and a counter increments every time a new, previously unseen substring appears. Finally, the counter is normalized by a theoretical upper bound, n divided by the base-2 logarithm of n, yielding a complexity value that ranges from 0 to 1 and is independent of sequence length.

The interpretation of this number is intuitive. A value close to 1 indicates a highly random, unpredictable signal with little internal self-similarity, while a value close to 0 indicates a repetitive, ordered series. Lempel-Ziv complexity has previously found applications in biomedical signal analysis, including studies of electroencephalogram background activity in Alzheimer’s disease patients and the detection of ventricular tachycardia and fibrillation in cardiac recordings. Applying it to nanopore data, however, allowed the researchers to ask a question that conventional event statistics cannot easily answer: how complex, in an information-theoretic sense, is the pattern of molecular blocking events?

The answer depended strongly on how much analyte was present. At the lower PEG concentration of 400 micromolar, complexity values were consistently lower than at 1000 micromolar, across every voltage tested. The full range of measured values stretched from a minimum of 0.45, recorded at 400 micromolar and +40 millivolts, to a maximum of 0.97, recorded at 1000 micromolar and +80 millivolts. The physical explanation is straightforward: higher analyte concentrations produce more frequent blocking events, filling the time series with rapid alternations between open and blocked states and pushing the complexity measure toward its upper limit. At lower concentrations, longer stretches of open-pore current dominate, and the signal becomes more ordered and less random.

To probe whether the apparent order at low concentration reflected genuine structure or simply sparse sampling, the researchers performed a clever control. They segmented the original time series recorded at 400 micromolar, shuffled the segments into a random order, and recalculated the complexity of the resulting randomized series. For every applied potential, the shuffled data yielded higher Lempel-Ziv values than the original recordings. This result is diagnostic: if the original series had already been maximally random, shuffling it would have changed nothing. The fact that randomization increased complexity means the unshuffled sequences contained residual structure, and that the biosensor’s output at low analyte concentration falls short of full stochastic behavior.

The team also compared series built exclusively from blocked periods with series built exclusively from unblocked periods, at both concentrations and all voltages. Statistical analysis using Student’s t-test, with a significance threshold of p less than 0.05, found no meaningful differences between the complexity of the blocked-only and unblocked-only series, nor between those segmented series and the original combined recordings. In other words, the complexity of the signal does not reside preferentially in either the moments when PEG occupies the pore or the intervals between occupations; the stochastic character is distributed across the entire record.

The broader implications reach into a long-standing debate in biology about the role of randomness. Molecular biology has traditionally leaned on deterministic principles, yet stochasticity is now recognized as a genuine feature of living systems rather than mere noise. Ion channel kinetics are widely treated as stochastic events, and some theorists argue that intrinsic randomness in the nervous system may even enable flexible decision-making. As the authors note, biological studies often lack the tools to distinguish chaotic, noisy, deterministic, and probabilistic behavior. Lempel-Ziv complexity offers one such tool. Their conclusion is nuanced: the alpha-hemolysin biosensor behaves in a purely stochastic fashion only at higher analyte concentrations, while at lower concentrations its stochasticity diminishes, though it can be mathematically restored by randomizing the event sequence. For engineers designing nanopore sensors, that finding matters, because the reliability of single-molecule detection depends on understanding when the digital current signature is truly random and when it carries hidden structure that smarter algorithms might exploit.

Subject of Research: Stochastic analysis of alpha-hemolysin nanopore ionic current blocking events using Lempel-Ziv complexity

Article Title: Analyzing alpha-hemolysin nanopore behavior using Lempel-Ziv complexity

Article References: Neves, G. F., Machado, D. C., Consoni, L. H. A., Costa, E. V. L., Carneiro, C. M. M., Rodrigues, C. G., & Nogueira, R. A. (2026). Analyzing alpha-hemolysin nanopore behavior using Lempel-Ziv complexity. Discover Chemistry, 3(1), Article 571. https://doi.org/10.1007/s44371-026-01022-8

Image Credits: AI Generated

DOI: 10.1007/s44371-026-01022-8

Keywords: alpha-hemolysin, nanopore, Lempel-Ziv complexity, stochastic sensing, polyethylene glycol, ion channel, planar lipid bilayer, biosensor, information theory, single-molecule detection, biophysics, stochasticity

Planning is one of the most computationally demanding things the human mind does. To decide on a course of action, people must mentally simulate future states of the world, evaluate the consequences of each possible move, and compare the expected values of different sequences of choices. In a perfectly predictable environment, this simulation can be carried out with confidence: if you take action A, outcome B will follow. But the real world is stochastic. Outcomes are corrupted by noise, the rules governing them shift over time, and sometimes our own actions fail to produce the effects we intend. Each of these forms of randomness, the authors note, makes exact planning exponentially harder, because the decision-maker must average over many possible futures rather than tracing a single one. The theoretical optimal solution to such problems quickly becomes intractable, which raises a fundamental question that has lingered in cognitive science: how do people balance the cognitive cost of planning against its potential benefits when the environment is uncertain?

To answer this question, the team designed a laboratory planning task in which participants encountered one of three distinct forms of stochasticity that are commonly found in real-world environments. The first was reliability, meaning the degree to which outcomes consistently reflect the underlying value of an action. The second was volatility, meaning the rate at which the environment’s structure changes over time. The third was controllability, meaning the extent to which a person’s own actions reliably determine what happens next. These three manipulations capture, in simplified form, the kinds of unpredictability people face when navigating unreliable transportation systems, volatile financial markets, or social situations where other people’s behavior only partially responds to our own. By varying each form of stochasticity independently, the researchers could ask whether different types of randomness affect planning in different ways, or whether the brain responds to all of them with a common strategy.

The central behavioral measure was deceptively simple: the time participants took to make their first choice in each planning episode. First-choice response times are widely used as a proxy for planning depth, because a longer deliberation typically reflects more mental simulation of future possibilities. Across all three manipulations, the researchers found a robust and consistent pattern. As stochasticity increased, people reduced their planning effort, as indexed by faster first-choice response times. In other words, whether the randomness came from unreliable outcomes, a shifting environment, or diminished control over events, the response was the same: participants spent less time deliberating. This consistency across three conceptually distinct sources of uncertainty is one of the study’s most important contributions, suggesting that the brain may treat different forms of environmental randomness through a common computational lens when deciding how much to think.

To understand what was happening inside the mind during these decisions, the team went beyond response times and built a family of computational cognitive models designed to account for participants’ choices. This modeling approach is a hallmark of the Ma laboratory’s work on resource-rational cognition, the idea that the brain optimizes not for perfect performance but for the best performance achievable given limited time, energy, and neural machinery. The models allowed the researchers to distinguish between competing hypotheses about how people cope with stochasticity. One possibility is that people perform something close to the correct calculation, computing expected values by averaging over possible outcomes weighted by their probabilities. Another possibility is that people adopt a much simpler strategy, one that sacrifices accuracy in exchange for a dramatic reduction in mental effort.

The modeling results pointed decisively toward the simpler strategy. Rather than calculating expected values optimally, people behaved as if the world were deterministic, a phenomenon the authors call determinizing. Under this scheme, the planner ignores the probability distribution over outcomes and plans as though each action leads to a single, certain result. Determinizing is computationally cheap: it converts a difficult problem of averaging over many futures into an easy problem of following one imagined path. The cost, of course, is accuracy, because in a genuinely stochastic world the single imagined path will sometimes be wrong. But the benefit is a substantial saving of cognitive effort, and the study suggests that when randomness rises, the brain judges that saving to be worth the loss in precision. This finding connects to a broader theme in recent cognitive science, in which people construct simplified mental representations to make planning tractable, rather than brute-forcing optimal solutions their neural hardware cannot afford.

Within this determinizing framework, the researchers identified a second, subtler signature of reduced effort. As stochasticity increased, people decreased their sensitivity to values, a pattern the authors describe as policy compression. In a fully sensitive policy, small differences in the value of different options translate into large differences in choice behavior, reflecting careful, fine-grained evaluation. When sensitivity to values decreases, choices become less discriminating, as though the planner is working from a coarser, more compressed summary of the decision landscape. Policy compression is a form of lossy compression of a behavioral policy: the brain retains the broad shape of what is worth doing but discards the fine distinctions that would require more computation to resolve. Consistent with the response time findings, this compression deepened as environmental randomness grew, providing converging evidence at both the level of deliberation time and the level of choice structure that planning effort was being actively withdrawn.

The study also carries a methodological warning for the field. The authors emphasize that their results reveal the limitations of studying stochasticity solely through single-shot decisions, the brief one-off gambles that dominate much of the decision-making literature. In a single-shot choice, the computational burden of planning is minimal, and the effects of stochasticity on effort may be invisible. It is only in tasks that require genuine multi-step planning, where the decision-maker must simulate sequences of future actions, that the withdrawal of effort under uncertainty becomes measurable. This suggests that conclusions about how humans handle uncertainty drawn from simple gambling paradigms may not generalize to the richer, sequential decisions that fill everyday life, from choosing a commute route to planning a career move. The planning task developed here offers a template for probing those richer decisions under controlled variation in environmental structure.

The broader implications of the work extend beyond the laboratory. If people systematically reduce planning effort as their environments become less reliable, more volatile, or less controllable, then the cognitive consequences of living in unstable conditions may be more nuanced than commonly assumed. It is tempting to interpret poor planning under uncertainty as a failure of attention or motivation, but this study reframes it as a potentially rational allocation of a scarce cognitive resource: when the expected payoff of careful planning falls because randomness will erode the value of any plan anyway, the brain sensibly stops paying for it. At the same time, the finding raises questions about contexts where reduced deliberation is harmful, such as volatile but high-stakes environments where careful planning remains valuable despite the noise. Understanding when determinizing serves us well and when it leads us astray is a natural next step. The research was funded by the National Science Foundation and the National Institutes of Health, and the authors acknowledge guidance from Marcelo Mattar, Todd Gureckis, and Cristina Savin. As stochasticity shapes everything from economic turbulence to climate variability, this work highlights an often overlooked truth: the mind’s response to an unpredictable world is not to think harder, but to think less.

Subject of Research: How environmental stochasticity affects human planning effort and strategy

Article Title: Environmental stochasticity reduces human planning effort

Article References: Lei, J., Olieslagers, J., Arfaei, N., Lin, D. X., & Ma, W. J. (2026). Environmental stochasticity reduces human planning effort. Nature Communications. https://doi.org/10.1038/s41467-026-78023-9

Image Credits: AI Generated

DOI: 10.1038/s41467-026-78023-9

Keywords: planning, stochasticity, decision-making, cognitive effort, determinizing, policy compression, computational modeling, response times, uncertainty, Nature Communications, resource rationality, human behavior

To disentangle the two kinds of uncertainty, the team designed a three-option probabilistic reversal-learning task framed as a fishing game, in which participants chose among three islands to maximize their catch. The researchers independently manipulated two dimensions: stochasticity, the random variability of outcomes under stable rules, and volatility, the rate at which the underlying reward contingencies actually changed. In the low-stochasticity environment, the best patch paid off on 80 percent of trials; in the high-stochasticity environment, it paid off only 60 percent of the time, making individual outcomes far less informative. Volatility was manipulated by reversing which patch was best every 25 trials in slow conditions and every 10 trials in fast ones. Because normative learning theory predicts that learning rates should rise under volatility but fall under high stochasticity, this design allowed the researchers to test whether anxious individuals violate these prescriptions in a selective way.

In the first experiment, fifty participants completed six 50-trial blocks spanning all combinations of stochasticity and volatility, and trait anxiety was measured with the State-Trait Anxiety Inventory. The results were unambiguous: participants with high trait anxiety, defined as a trait score of 45 or above, earned significantly fewer rewards than their low-anxiety peers, but only in environments that were stable yet noisy, that is, under slow volatility combined with medium or high stochasticity. When the environment genuinely changed rapidly, the two groups performed identically. This selectivity is the study’s central surprise. It rules out the idea that anxiety produces a blanket overestimation of environmental change; instead, anxious individuals remain sensitive to true volatility but fail to discount random outcome noise appropriately when the environment is actually stable.

Computational modeling revealed the mechanism behind this performance gap. The researchers fit a hierarchical Bayesian reinforcement-learning model with separate learning rates for rewards and punishments, allowing them to quantify how strongly each participant updated their beliefs after positive and negative feedback. Contrary to the widely held view that anxiety amplifies learning from negative outcomes, the elevated learning rates in high trait anxiety were reward-driven. Under slow volatility, higher trait anxiety predicted increasing reward learning rates as stochasticity rose, with 95 percent highest density intervals well above zero in both medium and high stochasticity conditions. Under fast volatility, reward learning rates did not vary with anxiety at all. Punishment learning rates, meanwhile, remained below task-optimal levels in both groups and did not differ between them, indicating that punishment sensitivity could not explain the performance differences.

To determine whether these elevated learning rates were actually harmful, the team simulated reinforcement-learning agents across nearly sixty thousand parameter combinations to identify the reward-maximizing learning rate for each environment. As normative theory predicts, the optimal learning rate increased with volatility and decreased with stochasticity. Against this benchmark, high trait anxious individuals significantly exceeded the optimal reward learning rate in the stable, noisy environments, with Bayes factors exceeding ten billion in the medium-stochasticity condition. Higher learning rates were in turn associated with lower reward accumulation, and anxious participants clustered in the region of high learning rates and low earnings. In other words, anxious learners were overfitting noise, updating their beliefs on the basis of random reward fluctuations as if each surprise signaled a genuine shift in the world.

The decision process itself was also altered. Using hierarchical drift-diffusion modeling, which decomposes reaction times into parameters such as drift rate, the speed of evidence accumulation, and boundary separation, the threshold of information required before committing to a choice, the researchers found that in the slow-volatility, high-stochasticity environment, high trait anxious participants showed drift rates centered near zero, dramatically slower than those of low-anxiety peers, along with elevated decision boundaries. Under fast volatility, none of these differences appeared. This means anxious individuals were not only updating their beliefs too aggressively but also accumulating evidence more slowly and cautiously when valuing the most rewarding option, a double computational burden confined precisely to environments that demand a clean separation of signal from noise.

The second experiment asked whether sleep, a physiological state tightly intertwined with anxiety, could modulate these computational phenotypes. Forty participants performed the task before and after sleeping in the laboratory, with overnight sleep recorded using a validated wearable EEG headband. Greater N3 sleep, the deepest stage of non-REM sleep characterized by slow delta oscillations and strongly linked to synaptic plasticity, was associated with a larger evening-to-morning reduction in state anxiety. Contrary to the researchers’ initial prediction, it was the high trait anxious participants who derived the greatest anxiolytic benefit, showing the steepest dissipation of morning state anxiety as their N3 duration increased. No comparable associations emerged for lighter N1 or N2 sleep or for REM sleep.

The restorative benefit of deep sleep extended directly into the realm of learning. Post-sleep reward accumulation was higher than pre-sleep performance overall, and greater N3 sleep specifically predicted higher post-sleep earnings and more accurate identification of the most rewarding patch in high trait anxious individuals, particularly under slow volatility. At the computational level, reward learning rates decreased from pre- to post-sleep under high stochasticity across all participants, consistent with normative adjustment, and greater N3 sleep was associated with significantly reduced post-sleep reward learning rates in the anxious group. When compared against the simulated optimal benchmarks, participants with more N3 sleep showed learning rates that were statistically indistinguishable from optimal values after sleep, whereas those with less N3 sleep continued to exceed the optimal benchmark. Post-sleep drift rates also rose, indicating faster, more confident evidence accumulation favoring the best option.

Taken together, the two experiments sketch a coherent picture in which trait anxiety selectively corrupts the inference that separates stochastic noise from genuine environmental change, inflating reward learning rates in stable but noisy settings and slowing the accumulation of decision evidence, while N3 sleep acts as a state-dependent corrective that nudges these parameters back toward their task-optimal values. The authors caution that N3 sleep was measured rather than experimentally manipulated, so its effects remain partly correlational, and that related constructs such as intolerance of uncertainty were not assessed, leaving open whether the observed effects are unique to trait anxiety. The sample was also drawn from a single campus community, and circadian factors were not controlled. Even so, because the trade-off between stochasticity and volatility is fundamental to countless real-world learning problems, from financial markets to social relationships, the study raises the provocative possibility that protecting deep sleep could be a tangible, physiological lever for improving how anxious minds learn under uncertainty.

Subject of Research: How trait anxiety and N3 sleep shape reward learning and decision-making under stochasticity and volatility

Article Title: Trait anxiety selectively amplifies reward learning under uncertainty, while N3 sleep attenuates it

Article References: Deshmukh, R., & Ramakrishnan, A. (2026). Trait anxiety selectively amplifies reward learning under uncertainty, while N3 sleep attenuates it. iScience, 29(10), Article 117745. https://doi.org/10.1016/j.isci.2026.117745

Image Credits: AI Generated

DOI: 10.1016/j.isci.2026.117745

Keywords: trait anxiety, reward learning, uncertainty, stochasticity, volatility, N3 sleep, reinforcement learning, drift-diffusion model, reversal learning, decision-making, sleep EEG, computational psychiatry