2.1.1. Metabolic considerations: The very idea

To understand metabolic considerations, let’s start with an example. How does the visual system code for decreases in local contrast? One option is to use decreasing firing rates from a high baseline. Maintaining that baseline, however, would use a lot of energy. By contrast, having dedicated “off” cells is metabolically feasible. Koch (Reference Koch2004, p. 477) uses this example to illustrate how “metabolic considerations” can constrain the style of computation. On our account, Koch’s example uses metabolism to evaluate which models of a computational task are biologically plausible (section 3.1).

The example illustrates that metabolic considerations involve reasoning from brain metabolism to models of cognitive capacities or neural systems. Brain metabolism includes processes that convert nutrients (e.g., glucose) into energy (e.g., ATP) and processes that remove neurotoxic byproducts of such energy conversion (Dienel, Reference Dienel, Byrne, Heidelberger and Waxham2014). Cognitive capacities are activities or dispositions that enable organisms to process information about bodily states and the sensory environment that is relevant to behavior (Bechtel, Reference Bechtel2008). Paradigmatic cognitive capacities include perception, memory, attention, and decision-making.

Models of cognitive capacities are idealized, abstract systems whose elements stand in a relation to assumed elements of the target capacity. Many models are mathematical and posit computational operations to explain cognitive capacities (Moustafa, Reference Moustafa2017). There are also dynamical models that use mathematical equations but eschew attributing computations to the brain. Mathematically, dynamical models are often compatible with the assumptions of information theory, as long as neither is interpreted as getting at the fundamental nature of the cognitive system (Beer & Williams, Reference Beer and Williams2015). Similarly, the account developed here focuses less on model ontology but instead highlights whether these models include metabolic considerations at all. Do modelers explicitly represent brain metabolism, constraining known models or generating new ones? Metabolic considerations also apply to models of neural systems, that is, systems of functionally and structurally connected neurons and other tissue, which are not of any one cognitive capacity.

Metabolism is traditionally backgrounded in computational modeling. From pioneering texts to many canonical approaches today, computational modelers proceed from the assumption that the brain is energy-efficient. They acknowledge that the brain is much more energy-efficient than supercomputers (Churchland & Seijnowski, Reference Churchland and Sejnowski1992; Stiefel & Coggan, Reference Stiefel and Coggan2023). Yet, when it comes to knowledge relevant for modeling cognitive capacities, most textbooks and overview articles background or ignore metabolism (Churchland & Seijnowski Reference Churchland and Sejnowski1992, pp. 50–59; Rieke et al., Reference Rieke, Warland, van Steveninck and Bialek1999; Dayan & Abbott, Reference Dayan and Abbott2005; Anderson, Reference Anderson2014; Kriegeskorte & Douglas, Reference Kriegeskorte and Douglas2018). The result is a modeling practice that “neglect[s] completely the energetic aspect of brain functioning, as if neural information processing were for free and performed in some abstract ‘mathematical’ hyperspace” (Karbowski, Reference Karbowski2024, p. 1).Footnote ¹

Against this widespread tendency, the concept of metabolic considerations foregrounds knowledge about metabolism when evaluating and generating models of cognitive capacities or neural systems. Modelers should not stop at qualitative descriptions of information processing as “expensive” or “cost-efficient.” As the next sections show, neuroscience now provides detailed quantitative knowledge that can be incorporated into cognitive modeling.

2.1.2. Efficiency of brain metabolism: No reason to background

The most generic way of including metabolism in cognitive modeling is assuming that the modeled system is efficient. “Efficiency” can be understood as a maximization of useful output units per unit of input. Yet, that brain metabolism is efficient can mean different things (Theriault et al., Reference Theriault, Shaffer, Dienel, Sander, Hooker, Dickerson, Barrett and Quigley2023, Table 1). Processes in the brain can maximize (1) ATP output per input unit (energy production efficiency), (2) energy consumption per second (energy consumption efficiency), (3) speed of signal transmission in meter/second (signal transmission efficiency), (4) information units sent per time unit (information rate efficiency), or (5) information units sent per ATP molecule (information-energy efficiency). When discussing how metabolic considerations matter to cognitive modeling, we must distinguish these different senses of “efficiency.”

Consider, for instance, (1) energy production efficiency, which arises from evolutionary pressures on brain organization. Organisms with less production-efficient brains must spend more time finding food, leaving them at a reproductive disadvantage (Pontzer, Reference Pontzer2015). If “efficiency” just meant global energy production efficiency via natural selection, then cognitive models could justifiably background metabolism. Brains would use energy efficiently because the organism as a whole has evolved to produce energy efficiently.

However, there is more to say about efficiency (Padamsey & Rochefort, Reference Padamsey and Rochefort2023). To begin, global energy production is just one application of production efficiency (1). While global glucose brain metabolism produces energy close to the theoretical maximum, its efficiency drops when organisms execute tasks (Dienel, Reference Dienel2019, p. 950). On the local level, however, brain metabolism is much less production-efficient. Alongside oxidative metabolism (∼30 ATP per glucose molecule) the brain continually uses glycolysis, which typically is thought of as a process which only occurs when oxygen supply is deficient. Yet, in the oxygen-rich brain this process occurs as aerobic glycolysis, which produces 2 ATP per glucose (section 3.2.2). Furthermore, action potentials are less (2) consumption-efficient but more (3) transmission-efficient than chemical diffusion (Sengupta Laughlin & Niven, Reference Sengupta, Laughlin and Niven2014). Finally, thick axons are more (4) rate-efficient than thin axons, whereas for (5) information-energy efficiency, thin axons outperform thick axons. This quick survey reveals that backgrounding the brain’s metabolism “because its efficient” hides potentially valuable information for cognitive modeling.

When models make efficiency considerations explicit, they often remain generic. Two prominent examples illustrate this point. One of the first computational models to incorporate metabolism found that model neurons do not maximize the number of distinct patterns they represent (Levy & Baxter, Reference Levy and Baxter1996). Rather, they maximize the amount of represented information per energy unit – they are information-energy-efficient. It does not matter for this model what the neuronal firing patterns represent. The finding is a generic statement about information-energy-efficient coding, not a specific constraint or guidance to model cognitive capacities.

The same is true for the role of efficiency in design principles of neural systems, such as “send only what is needed; send at the lowest acceptable rate; minimize wire, that is, length of all neural processes” (Sterling and Laughlin Reference Sterling and Laughlin2015, p. 54f., emphasis in original). These principles are derived from empirical regularities, such as the law of diminishing returns for the information rate in axons (Figure 1).Footnote ²

Figure 1.

The relation of the information rate in myelinated axons (y-axis) to their ability of producing energy in mitochondria (x-axis). If the axons get thicker than 0.7 μm, the information rate rises slower than the energy production rate. Reproduced from Sterling and Laughlin (Reference Sterling and Laughlin2015), Figure 3.6. The diagram is based on empirical data from Perge, Koch, Miller, Sterling, and Balasubramanian (Reference Perge, Koch, Miller, Sterling and Balasubramanian2009).

Design principles generically relate information rate efficiency and energy consumption. However, they cannot be directly applied to models of neural systems with specific information rates underlying cognitive capacities. To do so, modelers must specify what information is needed to execute that particular capacity, or what rate is acceptable for that specific system. Design principles provide a useful starting point but do not themselves constitute metabolic considerations needed for cognitive modeling.

2.1.3. Global energy budgets: Energy for cognition is limited

Global energy budgets are more specific than efficiency considerations, as they distinguish whether energy is spent for signaling versus non-signaling or “housekeeping functions” (Atwell & Laughlin, Reference Attwell and Laughlin2001). This distinction puts an upper bound on the energy that brains can allocate to cognitive capacities. Despite a lack of a mark of the cognitive (Allen, Reference Allen2017), neuroscientists and philosophers commonly characterize cognition as information processing relevant to the behavior of the organism (de Charms & Zador, Reference deCharms and Zador2000; Bechtel, Reference Bechtel2008; Haueis, Reference Haueis2018). Since only signaling functions are related to information processing, only that part of the energy budget can be used to fuel cognitive processes.Footnote ³ Figure 2 shows that recently, more direct estimates of non-signaling functions reduced the estimated energy available for information processing from 75% (Figure 2A) to 60% (Figure 2B). Energy budgets thus provide global metabolic constraints for cognitive modeling, since energy availability limits the amount of information processing brains can perform (section 3.1.1).

Figure 2.

Two recent energy budgets of signaling and non-signaling (“housekeeping”) functions in the brain (adapted from Howarth et al. Reference Howarth, Gleeson and Attwell2012, Figure. 1B and Engl and Atwell Reference Engl and Attwell2015, Figure 5).

Being a global upper bound, however, energy budgets provide no quantitative metabolic considerations to model specific cognitive capacities. Unless one is evaluating the energy cost of whole-brain models (Ritter et al., Reference Ritter, Schirner, McIntosh and Jirsa2013; Meier et al., Reference Meier, Perdikis, Blickensdörfer, Stefanovski, Liu, Maith, Dinkelbach, Baladron, Hamker and Ritter2022), these budgets provide limited guidance by estimating relative costs of neural activities. The budget in Figure 2A, for example, estimates that among signaling functions, generating action potentials is cheap (16%) compared to synaptic transmission (44%). So, in models of a specific cognitive capacity that include both processes, their energy costs should differ by a factor of 3. Nonetheless, global energy budgets are generic: they do not provide concrete energy costs for neural activities underlying specific cognitive capacities.

2.1.4. Numerical values for metabolic cost of information processing

Even more specific than energy budgets are numerical values of how much energy information processing tasks cost. These numerical values are already used to evaluate and generate models of cognitive capacities, such as memory, or specific computations, such as prediction error (sections 3.1.2 and 3.2.2). Such numbers have been calculated, for example, for the cost of transmitting information across synapses or axons (Harris, Jolivet, & Attwell, Reference Harris, Jolivet and Attwell2012; Sengupta et al., Reference Sengupta, Faisal, Laughlin and Niven2013), and for storing memories in synapses (Karbowski, Reference Karbowski2019; Karbowski & Urban, Reference Karbowski and Urban2024). These values are relevant for any model that includes neural information transmission and storage as a basis for executing cognitive capacities.

To estimate numerical values, researchers (1) obtain empirical or simulation data of the metabolic cost of a neural activity, (2) choose a theoretical measure of information processing, and (3) estimate the cost of 1 bit of information being processed by the activity (in ATP/bit). Consider first the cost of information transmission at the synapse (Harris et al., Reference Harris, Jolivet and Attwell2012, p. 762). It combines the (1) value of 164000 ATP molecules per synaptic vesicle release (Laughlin & Atwell, Reference Attwell and Laughlin2001) with the measure of (2) mutual information, which specifies how much the patterns of postsynaptic currents tell us about presynaptic action potential inputs.Footnote ⁴ The result is that (3) transmitting information across synapses costs 24000 ATP/bit.

Consider next the cost of storing memories, which is used to evaluate synaptic memory models (section 3.1.2). In synaptic models of memory, information putatively is stored in synapses, which are shaped by experience-dependent plasticity processes. Yet, how can synapses act as long-term memory storage sites if they constantly rebuild their structure? The molecular process of autophosphorylation provides a solution because it allows a system to store information despite molecular turnover (Lisman, Reference Lisman1985; Koch, Reference Koch2004, p. 455). Because synapses contain CaMKII molecules, which can autophosphorylate, they can act as long-term memory storage sites. In the model discussed in section 3.1.2, the metabolic cost of this process and another protein (PSD-95) during synaptic long-term potentiation was calculated at 31000–91000 ATP/min (Karbowski, Reference Karbowski2019). This amounts to step (1) of estimating the cost of information processing, but (2) information theoretic measures and (3) values for ATP/bit are missing in this study. In follow-up work using the same nonequilibrium thermodynamics approach, Karbowski and Urban (Reference Karbowski and Urban2024) specify (2) information gain as the theoretical measure.Footnote ⁵ As a result, they estimate (3) the cost of storing information for 10 synapses at 5 million ATP/bit.Footnote ⁶

Finally, there is the cost of information transmission in axons, which is used to generate new models of neural systems underlying specific computations in the brain (section 3.2.2). To estimate energy costs in model axons, Sengupta et al. (Reference Sengupta, Faisal, Laughlin and Niven2013) calculated trial-averaged ATP molecules/second for fueling the Na⁺/K⁺ pump that must be restored to transmit action potentials along the axon. To (2) measure information, they subtracted the noise entropy (average of probability distribution for each word occurring at any time) from the total entropy (information capacity of the spike train). To calculate the cost of information transmission, they (3) plotted the ATP/bit against axon diameter, which is shown in Figure 3.

Figure 3.

The relation between information-energy efficiency (bits/ATP) and axon diameter in single-compartment models with different ion channel densities (where 1 represents the channel densities of Hodgkin and Huxley’s original model of the giant squid axon). Adapted from Sengupta et al. (Reference Sengupta, Faisal, Laughlin and Niven2013), Figure 3B.

In Figure 3, axon model compartments on the blue line spend 71500 ATP/bit if they are 0.56 µm in diameter. This cost rises tenfold for 5.6 µm thick axons, showing that thin axons are more information energy-efficient (Theriault et al., Reference Theriault, Shaffer, Dienel, Sander, Hooker, Dickerson, Barrett and Quigley2023, Table 1).

2.1.5. Beyond generic uses of metabolism in cognitive modeling

The previous sections sort metabolic knowledge along a continuum of generic to specific when it comes to modeling cognitive capacities and neural systems. Backgrounding metabolism “because it’s efficient” is the most generic. The various senses of “efficiency” challenge that this attitude is appropriate. Even when specific senses of efficiency are addressed, the models usually are generic. They only specify certain kinds of codes as efficient or certain design principles that equally apply to many brain activities or systems. Energy budgets are more specific because they distinguish signaling and housekeeping functions, but they only put an upper energy limit on the sum of cognitive capacities. The most specific are numerical costs of specific information processing activities by particular neural entities. Yet, these numbers often are extrapolated from animal studies and are not specific to a neural system of cognitive significance. To wit: it is one thing to know the cost of sending a bit of information across the synapse. It is another to know the cost of computing the Laplacian convolved with the Gaussian in the early visual system (to use the classic example of Marr, Reference Marr1982, p. 337).

The account of metabolic considerations in section 3 shows how these limitations can be overcome once researchers use metabolic knowledge to constrain and generate models of cognitive capacities or neural systems. Rather than treating it as generic or mere background knowledge, metabolic considerations can and should play an active role in cognitive modeling. Due to disciplinary compartmentalization, however, metabolic knowledge from neurochemistry is often not related to information processing or computation. Thus, cognitive and computational neuroscientists may perceive this knowledge as “merely” metabolic or “noncognitive” and thus irrelevant (see section 2.2). Metabolic knowledge, even if relevant and quantitative, may be formatted in the wrong way to be useable in cognitive modeling projects. For example: while ATP/bit can be used to evaluate existing or generate new cognitive models, ATP/second cannot, as it is not directly related to information processing in cognitive models.

These issues highlight three desiderata for metabolic knowledge to act as full-blown considerations in cognitive models. First, this knowledge should be specific to a particular computational operation (e.g., contrast detection, computing prediction error), a cognitive capacity (e.g., early visual processing, memory), or neural system (e.g., hippocampus, dorsal visual stream). The relevance of the metabolic process to the computational operations underlying a cognitive capacity should also be made explicit (e.g., the costs of storing or transmitting information). Finally, metabolic knowledge should be legible, that is, its format needs to be compatible with how the cognitive model is formulated (Colaço, Reference Colaço2025).

2.2.1. General philosophical considerations of metabolism

There are agreed-upon points among the few philosophers who discuss metabolism and its relation to cognition. At its broadest, they address that, in our terms, the energy intensive but consumption-efficient nature of brains raises questions of how the brain’s energy use relates to the activities it performs. For example, in an introduction to the computational theory of mind, Piccinini and Colombo declare:

Likewise, philosophers emphasize the two tidbits mentioned in section 1. Take Eliasmith:

He also notes:

The metabolism-friendly philosophers highlight that energy use is a biological basis for cognition as “without energy, there is no thinking and no computation” (Thagard, Reference Thagard2022, p. 70). Likewise, they take brain metabolism and information processing to be “tightly intertwined,” as “in the brains of creatures like us, you cannot have one without the other” (Cao, Reference Cao2022, p. 506). These philosophers take metabolism to be fundamentally important for cognition and its study.

2.2.2. Metabolism: Tightly connected or enabling cognition?

Philosophers who defend the importance of metabolism debate how it relates to cognition. Some accounts challenge a sharp distinction between the two. Like Cao (Reference Cao2022), Godfrey-Smith argues that cognition and metabolism “are tightly connected in organisms like us,” and the “line between the “information processing” side of human brain activity and the metabolic side is porous” (Reference Godfrey-Smith2016, p. 493). On Godfrey-Smith’s account, all metabolic systems connected to the environment exhibit cognitive, or at least “proto-cognitive,” behaviors. For instance, they sense and respond to environmental stimuli in variable and plastic ways. This connection between metabolism and (proto-)cognition complicates any demarcation between the two. The connection also seems to challenge the feasibility of comparing biological and artificial systems, given their match in information processing but their grave mismatch in energy use (section 4.3).

Processes that seem “merely” metabolic might serve computational roles. For instance, functional hyperemia, that is, excessive blood flow from a metabolic standpoint, may have activity-dependent neuro-modulatory function and hence be computationally relevant (Moore & Cao, Reference Moore and Cao2008). This example suggests that we might reserve judgment about whether processes are “merely” metabolic or “properly” cognitive. Likewise, metabolism can help evaluate the functions ascribed to biological systems. Metabolism might “anchor” the information processing activities of the brain (Cao, Reference Cao2022, p. 506). Even if metabolism and cognition can be distinguished, their intertwined nature entails that the former constrains the latter (see section 3.1.1).

Metabolism might be necessary for cognitive phenomena, but unlike the mechanistic components often included in explanatory models. For instance, Klein (Reference Klein2018) identifies resources which are transformed during the modeled process, not individually important to the system in which they are embedded, and causally promiscuous in their interaction with many different components. Resources also cannot be easily substituted; their realization is relevant to the system. Mechanical parts, by contrast, persist over the timescale of a process, are individually important to the system, and are causally conservative in their interaction, and mechanistic models are indifferent to their composition (Klein, Reference Klein2018, Table 1). A toy example of this distinction is a car’s engine: a resource is gasoline, while a mechanical part is a check valve. Metabolism would equally count as a resource on this account (Klein, Reference Klein2022).

There are also etiological dimensions of metabolism. For instance, Godfrey-Smith’s account stems from his idea that metabolism, by achieving internal maintenance and coordination despite environmental variability, might be an evolutionary precursor of cognition. Likewise, metabolism arguably plays a role in major transitions in cognitive evolution, where resource efficiency seems to act as a main driver: “the lineage may not have gained any new cognitive functions at all. Rather it was able to do the same thing more efficiently” (Barron, Halina, & Klein, Reference Barron, Halina and Klein2023, p. 6). On this account, the evolution of brain structure and function is greatly informed by the use of metabolic resources.

2.2.3. From philosophical interests to cognitive modeling

The preceding shows that a growing minority of philosophers have taken a “metabolic turn,” arguing that metabolism is crucial for an adequate understanding of cognition.Footnote ⁷ The focus of these debates is whether “mere” metabolic and “proper” cognitive processes can be distinguished. Those who answer “no” argue that metabolism fundamentally shapes the evolutionary and developmental trajectory of the brain, and they appeal to neural design principles to show that computational processes and energy consumption are tightly related. These considerations speak against the tendency of most philosophers to treat metabolic considerations as mere support for cognition (Chirimuuta, Reference Chirimuuta2024, p. 92), subordinate “implementational-level” details, or background conditions. This tendency is motivated by multiple realizability of mental function (Polger & Shapiro, Reference Polger and Shapiro2016) and medium independence of computation (Piccinini, Reference Piccinini2020), even though these notions per se do not speak against including metabolism in cognitive models (section 4.2). By contrast, the claims in this section suggest that we cannot make sense of cognition in the absence of metabolism or confidently judge that a given process is “merely” metabolic. Furthermore, comparing behavior of biological and artificial systems without considering energy use can be misleading (see section 4.3).

Nonetheless, even philosophers who are interested in metabolic considerations do not detail how these considerations can be incorporated into cognitive modeling. These accounts justify why the mind sciences should care about metabolism, but they provide little concrete guidance for the practicing scientist. One potential reason for this lack is that philosophers largely address the topic of metabolism at a different level of analysis. At most, they generally consider theoretical lower bounds on energy requirements for physical computation, such as Anderson and Piccinini’s brief discussion of Landauer’s principle ((Reference Anderson and Piccinini2024) Anderson & Piccinini, Reference Anderson and Piccinini2024, p. 107). These generalist appeals are a first step, but they are insufficient for incorporating metabolic considerations into models.

Our account of metabolic considerations builds upon these philosophical insights by showing how metabolism can inform modeling practices. Because this account treats incorporation as a modeling problem, it is compatible with metabolic processes being an enabling condition for cognition or being proper cognitive factors. The ontological character of metabolic considerations is secondary to their role in evaluating and generating models of cognitive capacities or neural systems.

3.1.1. General characterization

Evaluative metabolic considerations address whether a model of a cognitive capacity or neural system is metabolically viable. They can favor a certain model while deeming other models metabolically implausible or impossible. Incorporating metabolic considerations thus can reveal computational deficiencies in the model.

Metabolism can function as evidence to prune or partition the space of possible models, for example, mechanistic models (Craver & Darden, Reference Craver and Darden2013, chs. 7–8). Models of cognitive mechanisms specify parts and operations that manipulate, store, or transmit information (Bechtel, Reference Bechtel2008; Kriegeskorte & Douglas, Reference Kriegeskorte and Douglas2018) or the computations that a neurocognitive mechanism performs (Koch, Reference Koch2004; Piccinini, Reference Piccinini2020). If a model represents a mechanism whose operations are inconsistent with evidence about brain metabolism or energy use, researchers should eliminate it from the space of possible models. As resources, metabolic considerations can likewise “serve to invalidate (or at least cast suspicion upon) certain types of mechanistic explanation” (Klein, Reference Klein2018, p. 8–9). This idea generalizes, as metabolism can be used to evaluate the plausibility of any model of cognitive capacities, mechanistic or otherwise (section 3.1.4).

Evaluative metabolic considerations also can provide explanatory constraints (Ross, Reference Ross2023). Metabolism constrains cognitive modeling because it contrasts possible with impossible computational operations that underlie cognitive capacities. Metabolism limits at which rates neural systems can process information or which computations are possible to implement (Sengupta & Stemmler, Reference Sengupta and Stemmler2014). Explanatory constrains are construed as external to the process they limit (Ross, Reference Ross2023). Information processing and computation in brains require energy, but the metabolic processes producing this constraining resource are generally not viewed as operative in the mechanisms that compute or process information (Levy & Baxter, Reference Levy and Baxter1996; Lennie, Reference Lennie2003; Sengupta et al., Reference Sengupta, Faisal, Laughlin and Niven2013, Reference Sengupta, Laughlin and Niven2014; Sterling & Laughlin, Reference Sterling and Laughlin2015; Engl & Atwell, Reference Engl and Attwell2015).

Like other explanatory constraints, metabolism is more fixed than other factors in cognitive modeling. The global energy availability is fixed by evolution, which favors brains that produce and use energy efficiently (section 2.1.2). Brain metabolism also does not vary substantially between task and rest (Clarke & Sokoloff, Reference Clarke and Sokoloff1999; Raichle Reference Raichle2015). And while short-term increases to local energy are possible, they require diverting energy from other regions (section 4.2). Further, as constraints, metabolic processes structure or guide information processing and computation, but they do not trigger it. In cognitive models, information processing is triggered by external inputs or internal information-bearing states of the system (de Charms & Zador, Reference deCharms and Zador2000; Bechtel, Reference Bechtel2008; Kriegeskorte & Douglas, Reference Kriegeskorte and Douglas2018).

Metabolic explanatory constraints are similar to but more specific than resource rational analyses. This approach uses metabolism as one type of resource to select “the best biologically feasible mind out of the infinite set of bounded rational minds” (Lieder & Griffiths, Reference Lieder and Griffiths2020, p. 4). Resource rational analysis is evaluative because it uses resources, limitations, constraints, and costs to improve cognitive modeling. However, one problem of this approach that it lacks an account that unifies metabolism and other computational or informational constraints not directly related to metabolism (Ma & Woodward, Reference Ma and Woodford2020; Theriault et al., Reference Theriault, Coleman, Feldman, Fridman, Sennesh, Barrett and Quigley2020). Our account avoids this difficulty by focusing directly on metabolic resources while not denying that a unified account can be given. It also distinguishes evaluative from generative roles of metabolism, which to our eyes is not included in resource rational analysis.

3.1.2. Core example: Evaluating mathematical models of synaptic memory

A clear example of evaluative metabolic considerations is the comparison between mathematical models of synaptic memory (Benna & Fusi, Reference Benna and Fusi2016; Karbowski, Reference Karbowski2019). These models often face issues in adequately balancing the different synaptic activities that putatively underwrite memory formation, consolidation, storage, and retrieval. A crucial problem these models must resolve is the “plasticity–rigidity dilemma,” that is, the problem that plastic synapses are good at storing new memories but bad at keeping old ones. One model put forward to solve this problem is a linear cascade model (Benna & Fusi, Reference Benna and Fusi2016), which represents memory storage as a chain of biochemical processes with different timescales.

The model represented in Figure 4a works analogously to a set of beakers that are filled with liquid, interconnected with tubes, and varying in diameter. The first beaker represents the strength of synaptic weight and is the most plastic one, while all others represent biochemical processes. The last beaker is connected to a reservoir that is always at equilibrium. Long-term potentiation is represented by adding liquid, while long-term depression is represented by removing liquid. Because of bidirectional interactions, a beaker’s liquid level tends to settle into equilibrium at the level of its nearest neighbors. This allows for flexibility of memory because “a single synaptic modification is remembered as long as the liquid levels remain significantly different from equilibrium” (Benna & Fusi, Reference Benna and Fusi2016, p. 1699).

Figure 4.

Linear cascade model of synaptic memory consolidation. Adapted from Benna and Fusi (Reference Benna and Fusi2016) Figure 1.

This cascade model allows that branches can split off or merge randomly from the initial linear chain model and shows that when properly tuned, both have the same memory performance. Therefore, both the linear and the complex branching models solve the plasticity–rigidity dilemma by storing the memory signal in fast variables first (e.g., synaptic strength) and then transferring it to increasingly slower variables (e.g., biochemical processes).

Though Karbowski also constructs a cascade model of synaptic learning and memory, he criticizes that almost all previous modeling “neglects the energetic aspect” of storing memories in synapses (Reference Karbowski2019, p. 1473). The crucial difference between them is Karbowski’s inclusion of cyclic motifs (Figure 5B, C).

Figure 5.

Noncyclical (A) and cyclical models (B, C) of synaptic memory consolidation. Adapted from Karbowski (Reference Karbowski2019, Figure 1).

Whereas the complex branching in Benna and Fusi’s model might incidentally contain cyclic motifs, Karbowski claims that they are necessary. In linear cascade models, as Benna and Fusi (Reference Benna and Fusi2016) demonstrate, all biochemical processes become balanced over time. Metabolically speaking, balance means that synapses do not exchange any energy with their environment. Karbowski argues:

While linear cascade models may be able to solve the plasticity–rigidity dilemma, they cannot solve the problem of thermal death because they assume that a synapse at equilibrium would be alive and capable of storing memories. Once metabolism is considered, this equilibrium state becomes unrealistic, as it would correspond to a dead synapse. Arguably, models of synaptic memory should avoid explaining how synapses keep their memories by appealing to states that no living synapse can enter. This is a deficiency of these sorts of models.

To solve this problem, cyclic motifs can be added to cascade models (Karbowski, Reference Karbowski2019). Cyclically connected biochemical processes with bidirectional interactions remain at a nonequilibrium steady state, which requires energy and thus prevents the system from thermal death. For synapses, the most important cyclical motifs are phosphorylation–dephosphorylation cycles of proteins, such as actin or AMPA receptors. These cycles are the energetically dominant factor during the initial phase of LTP, when a memory is learned. To prevent equilibrium, one factor in the cyclic motif must be much smaller than the others. Synapses fulfill this condition because phosphorylation is much cheaper than dephosphorylation.

When comparing these models of synaptic memory, metabolic considerations play an evidential and constraining role. The demand that synapses must keep exchanging metabolites with their environment prunes the space of possible models underlying synaptic memory. It excludes acyclical models that allow biochemical cascades to reach equilibrium while including cyclical models in which balance is prevented. This evidential function fulfills the first feature of explanatory constraints: it is impossible to store a memory in a biochemically balanced synapse, as the synapse would face thermal death.

These metabolic constraints are external to the parts and operations underlying synaptic memory consolidation. Figure 5B shows that ATP hydrolysis powers the transition from a substrate complex q₀ to an activated protein p ₊ (such as phosphorylation). But the arrow in that figure indicates that the process is external to the synaptic memory mechanism. ATP hydrolysis can power other processes, whose values remain fixed while the synapse consolidates a memory (Karbowski, Reference Karbowski2019, Table 1). These costs guide and structure the outcome of synaptic memory consolidation. For example, since the lifetime of a memory trace grows faster than its metabolic cost, longer memories are relatively cheaper than shorter ones (Karbowski, Reference Karbowski2019, p. 1483). The cost of memories structures the consolidation process because it favors longer memories.

In sum, metabolic considerations show that synaptic memory storage should not be modeled as a deviation from the equilibrium state but instead must be understood as a perturbation that increases energy dissipation from the nonequilibrium steady state. Cyclical models incorporate the constraint that biological systems must avoid thermal death, whereas linear cascade models do not. This makes the latter deficient in accounting for memory storage, and this deficiency ties to the model’s lack of fit with these metabolic considerations.

3.1.3. Supporting example: Evaluating neuronal model dynamics

While the core example shows how metabolic considerations may categorically evaluate cognitive models as possible or impossible, researchers often seek to gradually improve their models by adding evidential constraints. This section discusses such an example of gradually improving the dynamics of a model by including metabolic constraints.

The example is the inclusion of metabolism in a leaky integrate-and-fire (LIF) model (Jaras et al., Reference Jaras, Harada, Orchard, Maldonado and Vergara2021). A LIF model neuron generates an action potential once it integrates enough synaptic input to push the membrane potential beyond a fixed threshold. The membrane potential itself dissipates over time because of a leak conductance. After the action potential, a refractory period describes the time to restore the resting potential. Available energy influences the dynamic of this restoration process, among other adaptation behaviors. Yet, “the model and its consecutive extensions neglect metabolic dependence in their dynamic” (Jaras et al., Reference Jaras, Harada, Orchard, Maldonado and Vergara2021, p. 5250).

The energy-dependent LIF model, or EDLIF model, remedies this neglect. Crucially, spike dynamics are energy-dependent, as an ATP deficiency temporarily disables the Na²-/K⁺ pump after an action potential occurs. The result is slight depolarization, which makes stimulus causing another action potential more likely. The EDLIF model captures that ATP differences make a difference to the reset value of the membrane potential. When compared to the standard LIF model, EDLIF reproduces a greater dynamic range of action potentials in line with available evidence. It also reproduces a linear relationship between ATP consumption and firing known from earlier studies.Footnote ⁸

The EDLIF model incorporates metabolism as explanatory constraint. First, ATP concentration is relatively fixed because homeostatic mechanisms balance neuronal energy supply and consumption (Jaras et al., Reference Jaras, Harada, Orchard, Maldonado and Vergara2021, p. 5251). Second, ATP concentration structures neural dynamics because it influences the probability of generating an action potential, but it does not trigger action potentials.Footnote ⁹ Finally, the EDLIF model does not contrast possible with impossible outcomes, but rather gradually improves the existing, energetically unconstrained LIF model. Evaluating a model’s metabolic plausibility can thus be a gradual affair. That is, evaluative metabolic considerations can partition the space of possible models.

3.1.4. Guidelines how to use metabolism evaluatively

We can combine the general characterization with the details of the examples to formulate general guidelines for evaluative metabolic considerations in cognitive modeling (Box 1). We regard it as a strength if the guidelines strike modelers as sensible and general. If they are sensible, then they should hopefully lead to the widespread inclusion of metabolism in model evaluation. If they are general, then they should be equally applicable to models of different cognitive capacities, of different biological detail, of different species, etc.

At the center of evaluative metabolic considerations is the comparison of models in terms of their adequacy. To do so, researchers need to specify which models should be evaluated (E1). The common basis are models of the same cognitive capacity or the same neural system. In both cases, the models to be compared must be models of the same target. Ideally, there should be at least two models that differ by at least one component (an entity, operation, or organizational feature of the target system being modeled). With these models in hand, researchers should specify which metabolic processes or facts about energy consumption should constrain the models (E2). Researchers should choose metabolic processes or facts that are relevant to the cognitive capacity or neural system. This differs from case to case. Exemplary processes are oxidative phosphorylation, aerobic glycolysis, or fueling the Na²-/K⁺ pump, while exemplary facts are global energy budgets, the costs of synaptic transmission, or that metabolically active systems stay at nonequilibrium. As we have shown, many of these processes and facts have been uncovered, and they have been linked to paradigmatic cognitive capacities and neural systems.

In the next step, said metabolic factors are used to evaluate models and their components (E3). The relevance of the chosen processes or facts is determined by their utility for constraining the model. They should allow researchers to distinguish between metabolically possible or impossible models or components. This is exemplified by the cyclic and noncyclic motifs in synaptic learning models of the core example. Alternatively, they should allow researchers to assess the plausibility of models or components. This is exemplified by spike dynamics in LIF models of the supporting example. Importantly, this guideline does not entail that models that do not explicitly represent metabolic considerations are necessarily deficient. Models might adequately fit metabolic considerations even if they do not explicitly represent them. In our examples, deficiencies are revealed, but this not a guarantee. Nonetheless, researchers can determine whether a model fits these considerations via this guideline. For this reason, requiring evaluative metabolic considerations is not equivalent to demanding that models must always explicitly represent metabolism.

Through this evaluation, researchers should use metabolism to prune or partition the space of possible models (E4). Models of cognitive capacities or neural systems should be constrained by evidence about the elements of the target system. Evidence about metabolism and energy consumption can be used to rule out hypothesized structures of the elements that are impossible to implement in biological brains. This prunes the space of possible models underwriting a cognitive capacity. In addition, metabolic considerations can be used to assess the plausibility of a model. This amounts to a partition within the space of possible models into more or less plausible ones given available evidence. Finally, researchers should use these considerations to evaluate how energy availability and use structures and guides the flow of information in the system they target (E5). Even when assumed to be external to the computational operations they enable, metabolic constraints can limit the flow of information in cognitive systems. Researchers can therefore use metabolism to assess when the brain uses energetically expensive or cheap information channels (section 3.1.2) or distinguish between informational states that are energetically easy or difficult to undertake (section 4.1).

In sum, our account of evaluative metabolic considerations points out their role as explanatory constraints, which can be used to prune or partition the space of possible models. Although already used in current modeling practice, their use is not widespread. The benefit of our account is that it shows the logic of evaluative metabolic considerations, helping modelers to use them productively in the future.

3.2.1. General characterization

Generative metabolic considerations allow researchers to formulate or develop new cognitive models. Using metabolic knowledge to generate a model belongs to the discovery process because it populates the space of possible models. Incorporating metabolism during discovery thus has different roles than during model evaluation. This section develops and makes precise these generative roles.

Generating models is valuable when researchers want to explore a target system (Gelfert, Reference Gelfert2016, ch. 6). In this process, generative metabolic considerations function like a puzzle piece when exploring the relation between brain structure and the operations underlying cognitive capacities or neural systems. Metabolism links these two aspects because computations use energy, and structure determines how neural systems produce and consume energy. Models that include metabolism should allow researchers to explore the relation between brain structure and computation with greater detail.

Generative metabolic considerations can fulfill functions of exploratory modeling found across the physical and life sciences (Gelfert, Reference Gelfert2016; Haueis, Reference Haueis2022). They can provide a starting point for further inquiry, for example, into the relation between brain structure, metabolism, and information processing or computation. They can provide a proof-of-concept that metabolic knowledge is relevant to modeling a particular cognitive capacity. They can deliver potential explanations, for example, of how a particular type of computation is implemented in a neural system. Lastly, they can be used to reassess the target system, that is, they can change assumptions about what kind of system the model targets.

3.2.2. Core example: Exploring efficiency tradeoffs in neural systems

A clear case of a generative use of metabolic considerations is from Theriault et al. (Reference Theriault, Shaffer, Dienel, Sander, Hooker, Dickerson, Barrett and Quigley2023), who use the metabolic process of aerobic glycolysis to develop the efficiency tradeoff hypothesis. This hypothesis links information processing in thin axons (structure) to the transmission of prediction errors (computation). During tasks and the experimental resting state, the brain produces energy both by oxidative phosphorylation in mitochondria (30 ATP per glucose molecule) and aerobic glycolysis, a process where glucose is converted to energy while oxygen is present (2 ATP per glucose molecule). This fact presents a puzzle: why does the brain meet increased energy demand with the production inefficient process of aerobic glycolysis, when it could use the more production-efficient process of oxidative phosphorylation?

The efficiency-tradeoff hypothesis is a possible solution to this puzzle. It first uses principles of neural design (send at lowest rate, minimize wire) to state that thin axons are information-energy-efficient. Thin axons fire less often (spikes/sec), making each spike more informationally rich (bits/spike). Their small diameter makes them cheaper to repolarize (Na+ ions have greater effects on membrane voltage). Taken together, this means that thin axons maximize bits/ATP. By doing so, however, axons become too thin to house the mitochondria needed for oxidative phosphorylation. Therefore, this production-efficient process cannot fuel firing in thin axons, particularly during peak energy demand. The solution to this problem is aerobic glycolysis, which does not require mitochondria and thus can fuel spike transmission in thin axons. This solution pairs with the efficiency tradeoff because a production-inefficient process (aerobic glycolysis) is used to fuel a process that maximizes bits sent per ATP used.

The efficiency-tradeoff hypothesis motivates a larger attempt to develop a cognitive model using a predictive coding framework, where prediction errors are computations of deviations from an internal model constructed by the system in question.Footnote ¹⁰ Figure 6 shows that metabolic considerations function like a puzzle piece that links aspects of brain structure to measurable quantities and to computational capacities of the brain.

Figure 6.

Theriault et al.’s conceptual model linking different features of neural structure and brain metabolism to measurable quantities and computational processes in the brain. Adapted from Theriault et al. (Reference Theriault, Shaffer, Dienel, Sander, Hooker, Dickerson, Barrett and Quigley2023), Figure 6.

The link to prediction error brings out how generative uses of metabolism contribute to modeling cognitive capacities and neural systems. Figure 6 links prediction error, among other things, to firing in thin axons, since the firing characteristics of the latter match the computational profile of the former. In this model, aerobic glycolysis fuels the forwarding of prediction errors via information-energy-efficient thin axons.

The efficiency-tradeoff hypothesis and subsequent model formulation showcase exploratory functions of metabolism in cognitive modeling. First, Figure 6 represents a starting point for further inquiry. It suggests experiments for testing the decoupling between glucose and oxygen metabolism and gamma oscillations (Theriault et al., Reference Theriault, Shaffer, Dienel, Sander, Hooker, Dickerson, Barrett and Quigley2023, 16). Second, the link between aerobic glycolysis and prediction error provides a proof-of-concept that metabolic considerations are relevant to modeling. Aerobic glycolysis is used in executive function (Madsen et al., Reference Madsen, Hasselbalch, Hagemann, Olsen, Bülow, Holm, Wildschiødtz, Paulson and Lassen1995), visual and tactile processing (Bednařík et al., Reference Bednařík, Tkáč, Giove, DiNuzzo, Deelchand, Emir, Eberly and Mangia2015; Díaz-García et al., Reference Díaz-García, Mongeon, Lahmann, Koveal, Zucker and Yellen2017), and spatial memory acquisition (Harris et al., Reference Harris, Lone, Lim, Martinez, Frame, Scholl and Cumming2019).Footnote ¹¹ This suggests that the efficiency-tradeoff hypothesis can be tested by experiments with tasks that operationalize these cognitive capacities, measurement tools that record activity in thin axons, and models that explain their activity in terms of prediction error transmission.

Third, the efficiency trade-off hypothesis provides a potential explanation to the puzzle of stimulation-based aerobic glycolysis. The existence of a tradeoff between two senses of efficiency (energy production and information rate efficiency), if corroborated, would explain why the brain uses an inefficient way to produce ATP when energy demand is high. Fourth, Figure 6 suggests that researchers should reassess the target system they are measuring, from one in which BOLD intensity increases indicate feature detection, to one where they indicate prediction error processing (cf., Theriault et al., Reference Theriault, Shaffer, Dienel, Sander, Hooker, Dickerson, Barrett and Quigley2023, pp. 23–24).

We analyze this example in detail not because we support every facet of this model. Rather, it shows that generative uses of metabolism occur. The core example shows how a model can be developed and adapted based on metabolic considerations.

3.2.3. Supporting example: Exploring energy-efficient learning strategies

Whereas the core example shows how metabolic considerations link known aspects of target systems, researchers can also use them to generate novel strategies of how neural systems execute cognitive capacities. An example is how energy efficiency can shape artificial neural network models of learning (Pache & Rossum, Reference Pache and Van Rossum2023). Assuming that animals have evolved and developed to learn by using as little energy as possible (section 2.1.2), Pache and Rossum explored energy-saving strategies in learning in neural networks trained with a backpropagation algorithm and tested on the MNIST digit classification task. In doing so, they find that “incorporating metabolic demands of learning leads to modifications of learning rules that are in line with experimental data” (Pache & Rossum, Reference Pache and Van Rossum2023, p. 5).

This exploratory study resulted in posits of three energy-saving strategies. First, a system saves energy by not updating on correct classifications. The saving scales with the size of the training dataset, though this increased consumption efficiency pairs with decreased robustness of the network to input variation. Second, a system saves energy by only updating synapses that require the largest magnitude of change. The saving scales with the number of synapses available given the relevant complexity of the task: if a network is far more complex than what is needed for the task, the savings are greater. Third, a system saves energy by using multiple forms of plasticity, allowing for some information to be cached in temporary, transient forms and other information to be consolidated into comparatively permanent forms. Unlike the others, this strategy is informed by higher energy costs of persistent, consolidated synaptic forms. The saving scales with the quantity of information updates that can be temporarily stored and minimally consolidated, though this must be balanced with potential information loss.

This work suggests learning rules that would be hard if not impossible to discover if modelers would ignore the brain’s need to save energy. It thus explores a topic rarely addressed in cognitive models: the metabolic costs of plasticity (see Gershman, Reference Gershman2023 for an exception). The example thus illustrates our view that generative metabolic considerations can populate the space of possible cognitive models.

3.2.4. Guidelines: How to use metabolism to generate cognitive models

We can combine the general characterization with the details of the examples to formulate general guidelines for generative metabolic considerations in cognitive modeling (Box 2).

As this strategy involves generating a model, what one aims to model must be specified. The starting point of these guidelines is positing or determining a cognitive capacity or target system to be modeled (G1). Typically, the choice will be between a cognitive capacity and a neural system that might underwrite multiple cognitive capacities. One might, for instance, focus on modeling the activities of the hippocampus, even if one does not intend to model solely memory formation or consolidation. What is key is that the target choice delimits all subsequent modeling choices and therefore the success of the strategy. Lack of precision in specifying the target can make subsequent choices more nebulous. With a capacity or a system specified, one should select some set of considerations that will inform the parameters of the model in question. This means that researchers should specify metabolic considerations that can inform one or more parameters in a model of the target (G2). Examples include specific details of the efficiency of brain metabolism, principles of neural design, the characteristics of energy budgets, and metabolic costs of information processing (section 2.1). These considerations should be made at the start of generative modeling and should build on established knowledge of metabolic considerations relevant to the modeled capacity or system.

With metabolic considerations in hand and a target chosen, researchers should either construct parameters that fit with these considerations or adapt parameters from existing models of the same or comparable targets (G3). Researchers should thus generate the model by constructing other parameters. Importantly, G3 accommodates the fact that these choices need not be made ex nihilo. Rather, researchers can adapt parameters from existing models and established metabolic considerations, drawing upon the specifications made in G2. This is exemplified by the adaptation of a predictive coding framework in the core example and by the adaptation of neural network architectures in the supporting example. This integration is facilitated by drawing on previous attempts to model the target in question.

Finally, researchers should explore variations in the chosen parameters and their effects on other parameters in the model, comparing and assessing these variations’ effects on the measure or manipulation of the chosen target (G4). One should explore how variations in parameters, including but not limited to those parameters informed by metabolic considerations, inform variations in other system parameters. In this sense, generative metabolic considerations provide: (1) a starting point for inquiry into the relation between brain structure and information processing, (2) a proof-of-concept that metabolic knowledge is relevant to cognitive modeling, or (3) a potential explanation of how a computation is implemented. This exploration can inform new hypotheses to be tested empirically and explored computationally or via simulation.

This final generative guideline shows that the generation and evaluation of models that incorporate metabolic considerations can form a virtuous cycle (Figure 7). The former populates a space of possible models for cognitive capacities and neural systems. That is, generation gives us new models. Evaluation prunes or partitions this space. It allows us to assess the relative strengths and deficiencies of different models, and it also helps us determine which models are possible or plausible. This virtuous cycle, which entails generating and evaluating cognitive models via metabolism, can be iterative. Metabolic considerations provide a framework for taking in new metabolic knowledge, and assessing this knowledge reshapes how we assess existing models of cognitive capacities.

Figure 7.

Schematic of the space of possible models for a cognitive capacity or neural system. Evaluative metabolic considerations prune the space by excluding models because they violate known metabolic constraints and partition the space by considering which models are more or less metabolically plausible. Generative metabolic considerations populate the space with new possible models, which can in turn be evaluated in terms of their plausibility.