Neuronal ATP dynamics.

In the EDLIF model, the ATP dynamics are characterized by considering two processes: those that supply ATP to the neuron (A_s(t)), and those that consume ATP (C(t)). The available ATP in the neuron (A(t)) can be formalized as follows [11]:

(6)

and the neuron’s homeostatic mechanisms attempt to keep ATP levels close to the homeostatic ATP level (A_H). The ATP production dynamics is mathematically formalized by Eq 7. Thus, ATP production depends on the actual ATP level with respect to a homeostatic one (K(A_H−A(t))) and a basal production term (A_B) accounting for resting potential and housekeeping activities:

(7)

where the parameter K ((1/ms) units) is the rate at which ATP is produced.

Regarding energy consumption, we consider four energy-consuming activities in this study. Namely, the energy related to the neuron’s housekeeping activities E_hk, the energy associated with maintaining the resting potential E_rp, the total energy expended by the neuron’s action potentials E_ap, and the total energy associated with receiving postsynaptic potentials due to a presynaptic neighbor neurons having action potentials (E_syn). Both rate consumptions E_hk and E_rp are measured in energy per time units. On the other hand, both consumption activities E_ap and E_syn refer to the total amount of energy consumed by those activities. To obtain the corresponding rate consumption for E_ap and E_syn we use temporal kernels. In this manner, the amount of energy expended per time unit associated with action potentials (or postsynaptic potentials) is described as (or ), where is the kernel associated to the process x and .

Originally, in the EDLIF model the production and consumption of ATP is measured in mM/ms, while the available ATP A(t) is measured in mM. However, experimentally, it is difficult to precisely measure ATP concentrations in single neurons or populations. To overcome this difficulty and to define a more general framework, here we will use percentage units to quantify available energy (A(t)), homeostatic energy (), and activity-related energy consumption such as action potential energy consumption (E_ap), house-keeping (E_hk), resting potential (E_rp), and postsynaptic energy consumption (E_syn).

Static synapses and energy consumption in the neuron.

To understand in detail the energy perturbations induced by presynaptic neurons in postsynaptic energy levels through synapses, let us analyze the neuron’s energy dynamics when several presynaptic neurons are connected to it. In a network, each neuron’s energy evolves following Eq 6, where , and the energy consumption C can be divided into the following terms:

(8)

where C_B represents the basal energy consumption associated with maintaining the resting potential and general housekeeping activities (which is equal to the basal production A_B in this work). C_ap is the energy consumption accounting for the neuron action potentials, and C_syn is the energy consumption related to receiving action potentials from other neurons through the synapses. Please realize that C_ap(t) (C_syn(t)) is different than E_ap (E_syn). The former is the action potential (postsynaptic) energy consumption through time, and the latter is the total amount of energy related to that activity. Thus, C_ap (C_syn) is obtained using E_ap (E_syn) and a kernel function , which describes how the total energy consumption is expended through time.

The solution to Eq 6 with an arbitrary energy consumption C(t) is described by Eq 9:

(9)

where is the rate at which ATP can be produced in the neuron (see Eq 7).

For one presynaptic action potential propagating through a synapse with fixed strength w, the synaptic consumption on the postsynaptic neuron can be described by:

(10)

where denotes the Heaviside step function, E_syn is the total energy expended for an incoming presynaptic action potential, is the absolute value of the normalized synaptic strength (, where w_max is the maximum allowed weight strength) and is a kernel describing how the synaptic energy expenditure is expended through time (). If we choose a kernel that can be linearly summed up, then the synaptic energy expenditure in the postsynaptic neuron due to several presynaptic cells with their respective spike times is:

(11)

Similarly, for each action potential in the postsynaptic neuron, the energy expenditure can be described by:

(12)

where E_ap is the energy expenditure related to one action potential and is analogous to , but for the postsynaptic action potentials (). If can be linearly summed up, then Eq 12 reads:

(13)

To find closed-form expressions for the available energy in the postsynaptic neuron, it is possible to define and kernels and plugin Eqs 11 and 13 into Eq 6. In particular, if exponential kernels are used for and , Eq 11 reads:

(14)

where is the energy time constant of the kth synapse and describes how fast is the change in the postsynaptic neuron’s energy given the incoming action potential through the kth synapse. Note that if the total energy consumption in the postsynaptic neuron produced by the presynaptic neuron sending one action potential through synapse kth is . Thus, Eq 14 guarantees that for the maximum allowed synaptic weight, the total energy consumption perceived by the postsynaptic neuron is exactly and a fraction () of otherwise. Similarly, using an exponential kernel, Eq 13 can be expressed as:

(15)

where is the time constant defining how fast is E_ap expended through time. Eq 15 guaranties that the neuron will consume a total amount of E_ap for each action potential.

Therefore, plugin Eqs 15 and 14 into Eq 9, we have a closed-form expression for the neuron’s energy dynamics under exponential kernels ( and ) for energy consumption:

(16)

Eq 16 describes the energy dynamics of a single neuron within a network, given all incoming synapses strengths and spike-times. Thus, it allows to precisely quantify and analyze the energy dynamics of each neuron within a network with arbitrary architecture.

Energy dependent Spike-timing-dependent plasticity model.

Here we leverage previously defined plasticity models to introduce an energy-dependent synaptic plasticity rule.

Particularly, we introduced an energy-dependent plasticity rule for synaptic strength. We follow the formalization of STDP introduced in [26]:

(17)

where the synaptic efficacy w is normalized to [0,1]. The learning rate () scales the magnitude of the individual weight change, is a temporal filter , and is defined as follows:

(18)

with accounting for the asymmetry between the scales of potentiation and depression.

In general, neurons harbor energetic/metabolic sensors such as AMP-activated protein kinase, which tends to restore ATP concentration by decreasing energy consumption while increasing energy production after different energetic challenges [27–29]. This same mechanism has been linked to plasticity [29], where they were able to show that energetic stress can suppress LTP, while treatments targeting energetic pathways can rescue it by removing the energetic stress. This findings inspired our LTP energy-dependent plasticity rule. Specifically, when glycolysis is pharmacologically inhibited (thus ATP production from glucose is inhibited), LTP is suppressed [29]. Therefore, our energy-dependent STDP rule should suppress LTP when there is a low energy level. Following this observation, we proceed to extend Eq 17 to account for energetics, while keeping in mind that this is a practical simplification of a more complex biological phenomena, but which allows deepening the understanding of metabolism and plasticity interactions, and its impact on neural network dynamics.

To account for energetics in STDP, it is possible to extend f₊(w) in Eq 18 to include the energy level A of the postsynaptic neuron, obtaining f₊(w,A). There are several ways to mathematically formalize the inclusion of postsynaptic energy level in f₊(w), but we follow a similar rationale as the one used in EDLIF neuron, namely, we want energy-dependent STDP to be sensitive to energy imbalance. Hence, we can investigate not only how energy affects STPD, but also how energy imbalance affects STDP given a certain level of energy imbalance sensitivity in that synapse’s plasticity. Using the rationale mentioned above we extend f₊(w) to f₊(w,A) as follows:

(19)

where is the sensitivity of the synapse’s plasticity to energy imbalances, A_H is the homeostatic energy level, and A is the postsynaptic energy level. STDP effects can be thought as a change in postsynaptic receptor density. Given that the postsynaptic neuron has to modify its receptor’s density, their energy level is affecting receptor density modifications. Please note that if the synapse is not sensitive to energy imbalances (i.e., ), then we recover the original STDP rule.

The postsynaptic energy level can be interpreted as a plasticity moderator. Thus, our energy-dependent STDP rule is a special case of a three-factor learning rule, with postsynaptic available energy A as modulator.

For the depression updating function f₋(w) we keep the original formulation (, see Eq 18) because we don’t have conclusive evidence justifying the depression’s dependence on postsynaptic available energy. However, energy may be also affecting depression.

Plugging Eq 19 into Eq 17, it is possible to describe weight’s update accounting for pre- and postsynaptic spike-time and postsynaptic energy level:

(20)

In principle, Eq 20 allows to include weight w dependencies in the updating rule if . In those cases, the updating rule is called a multiplicative rule, while if the updating rule is additive, because weight update is independent of the current weight value. Multiplicative rules generate uni-modal weight distributions, while additive rules generate bi-modal weight distributions [26]. Bi-modal weights distributions are observed in biology, and because the mathematical tractability of STDP additive rules is easier than multiplicative ones, in what follows we decided to focus on the additive update rule for weights update (). However, there is also significant biological evidence supporting multiplicative STDP as well. For this reason, we define Eq 20 in a general form, allowing the exploration of multiplicative energy-dependent STDP in future works.

Network’s firing-rate.

To improve our analytical understanding of the E-I network, it is practical to have an approximation of the network’s firing rate . In principle, using the LIF model (or the EDLIF model with no energy sensitivity to energy imbalances, i.e., ) and knowing the model’s parameters, the firing rate of a neuron excited by a constant current can be easily calculated. In a more general way, the firing rate of a neuron can be calculated by defining a function , mapping stimulation current to firing rate. If the neuron is insensitive to energy imbalance (i.e., ) and is excited with a supra-threshold current I, then the function can be found utilizing Eq 21:

(21)

where is the neuron’s refractory period, is the neuron’s time constant, and . In this case, , where I is the constant incoming current to the neuron. Thus, the firing rate of each neuron under a constant current stimulation can be calculated knowing the neuron’s parameters as well as the incoming constant current and the function, which is monotonically increasing in I. Therefore, if we can approximate the mean incoming current to a neuron in a small time-window, it is possible to approximate the neuron’s firing rate , where . Hence, to approximate each neuron’s firing rate, let us approximate the mean incoming current to each neuron:

(22)

Therefore, the mean incoming current to a neuron is the mean incoming synaptic current, plus the constant external incoming stimulation current injected into the neuron. Eq 22 describes the mean incoming current to a neuron utilizing an interaction kernel. If we use a delta Dirac interaction kernel (i.e., ), Eq 22 reads:

(23)

If we define the constant incoming stimulation current to each neuron as the dimension vector , and the mean incoming synaptic current as the dimension vector , Eq 23 can be expressed in matrix form as follows:

(24)

It is possible to apply point-wise to obtain the approximated network’s firing rate:

(25)

Eq 25 imposes a current-frequency constraint on the network. In general, defines a nonlinear relationship between incoming current and the neuron’s firing rate. However, we can Taylor expand Eq 25 around :

(26)

where is the identity matrix. Moreover, if is a linear transformation such as , the following relationship holds:

(27)

Eqs 26 and 27 shows that, if the weights are static on average (), then the firing rate of the network converges towards an equilibrium. This observation is important because it highlights the fact that, in general, constant weights are needed to achieve a constant firing rate.

Eq 25 defines a relation determining plausible weight-frequency values, which in principle, are independent of the network’s energy dynamics. In the Results section, we will explore how these constraints interact with the metabolic constraints at the network level.

Simulation details.

To numerically test our theoretical predictions, the network is simulated utilizing the Neural Simulation Tool program (NEST) [36] and the EDLIF neuronal model as well as the ED-STDP synaptic model are specified using NESTML [37], the domain-specific language tailored for the spiking neural network simulator NEST. Models and code are available on GitHub (https://github.com/Wiss/edsnn).

The simulated network has n = 500 neurons, and the excitatory-inhibitory ratio is , following biologically realistic excitatory-inhibitory ratio values [38].

Given the need for a mathematically tractable framework, the network architecture is defined with an all-to-all structural connectivity. In addition, following in vitro measured weights strength in neuronal cell assemblies [39], initial synaptic strength values for the simulated network are drawn from an exponential distribution (see Eq 28)

(28)

where is the probability density function of the exponential distribution. This means that the distribution of the weights w is described by an exponential distribution with scale parameter . Therefore, the expected value of the weights is and the variance . For the simulation we used .

To emulate the incoming activity to each neuron from other brain areas, we inject into each neuron a constant current drawn from a Gaussian distribution with mean and standard deviation . By assuming that all neurons share the same parameters values we study the homogeneous parameter scenario (neuronal parameters are in S1 Table).

Energy dynamic and fixed point analysis.

When studying the network’s dynamics, we are particularly interested in knowing if the network converges or oscillates toward a specific state. If the network evolves toward an specific state independently of the initial conditions, those states are attractors of the network. Therefore, to find fixed points in the network dynamics, we start by trying to find if there are energy attractors in the network dynamics, when all the neurons share the same parameters (the homogeneous case). Accordingly, we analyze stable fixed points for energy states of an arbitrary neuron in the network.

The total energy change of any neuron in the network in a time interval T is calculated by summing the contributions of energy production and consumption occurring in the time interval . From Eq 6 we obtain:

(34)

Previously, we formalized the synaptic energy consumption C_syn (see Eq 11) as well as the neuron’s own action potential energy consumption C_ap (see Eq 13), for arbitrary and kernels. To simplify the mathematical tractability, now we will choose the delta kernel for and . Thus, Eq 34 reads:

(35)

Because weights change slowly (i.e., in Eq 20), for small T we can assume they are constant in the interval (i.e., ). Formally, we are assuming that time scales of learning and neuronal spike dynamics can be separated [40]. In addition, because of the linearity of the integral operator:

(36)

where is the mean energy level of the neuron in the interval, and in the second and third terms we obtain the neuron self and connected neighbor’s firing rates, respectively. Now, if we impose in Eq 36, it is possible to find neuron’s firing rates, incoming weight, and energy expenditure values that satisfy the energy fixed point constraints:

(37)

which in matrix form reads:

(38)

where . Eq 37 holds if the neuron is in an energy fixed point and, to achieve the energy fixed point, the excitatory-excitatory connections to the neuron under study also need to achieve a fixed point (the other connections are static). Otherwise, the synaptic energy consumption (C_syn) will stay varying, impeding to achieve an energy fixed point in the postsynaptic excitatory neuron. Please realize that, if the excitatory-excitatory connections achieve a fixed point as well as the presynaptic neuron’s firing rate, then the firing rate of the postsynaptic neuron also achieves a fixed point (i.e., ).

Excitatory-excitatory connections stabilize at a fixed point when the postsynaptic excitatory neuron’s energy level matches . However, for silent neurons incoming weights remain unchanged, leading to a constant energy level in the neuron (), regardless of whether . Therefore, silent neurons (neurons with firing rate equal to zero) conform to Eq 37. If there is no rise in their incoming current, they will stay at the fixed point where , even without needing to meet the condition. Thus, for non-silent neurons in the fixed point, we can approximate , obtaining:

(39)

By rearranging Eq 39, we arrive to the following relationship constraining weights and firing rates of each neuron in the network:

(40)

If all the neurons in the network converge towards a fixed point, they must satisfy Eq 37 and, in particular, non-silent excitatory neurons must satisfy Eq 40. In consequence, in this state, all the excitatory-excitatory connections in the network remain constant (i.e., ) in average as well as the firing rates (i.e., ). As a result, Eq 40 allow us to predict the energy consumption of each non-silent excitatory neuron in the network after converging towards a fixed point.

The energy consumption induced by action potentials can be expressed in terms of the energy consumption induced by synapses i.e. (experimental findings indicate that m is approximately 1/3 [41]). Thus, Eq 40 can be rewritten as:

(41)

Consequently,

(42)

Eq 37 reveals a metabolic constraint over incoming synapse strengths and their respective presynaptic neuronal firing rates, affecting non-silent excitatory neurons in the energy fixed point. In particular, Eq 40 shows that, to achieve a metabolic fixed point in the excitatory postsynaptic neurons, there is a trade-off between synaptic strength and the corresponding presynaptic neuronal firing rate. Interestingly, the metabolic constraint in Eq 40 dictates an inverse relationship between weights and firing rates, which is a completely new constraint in the system, given that previous models generally do not account for metabolic activity. Moreover, the constraint mentioned above enables the emergence of an attractor state to the network, as a consequence of the interception between the metabolic and the physically plausible states of the network is given by the previously introduced metabolic (see Eq 40) and weight-frequency relations (see Eq 25).

On the one hand, Eq 25 gives a weight-rate relationship where increasing excitatory-excitatory weights generates higher excitatory firing rates , due to higher mean incoming synaptic currents to postsynaptic neurons. On the other hand, if either excitatory-excitatory weights or excitatory firing rates increase, the postsynaptic energy levels drop (see Eq 38). In addition, at the metabolic fixed point and under previously described assumptions, non-silent excitatory neurons fulfill Eq 42. As a consequence, if excitatory firing rates (excitatory-excitatory weight) increase, then excitatory-excitatory weights (excitatory firing rates) must decrease. Thus, Eq 42 imposes an inverse relationship between excitatory-excitatory weights and excitatory firing rates, contrary to Eq 25, where the relationship between the two variables is direct. Fig 4 shows a conceptualization of the previous reasoning.

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Fig 4. Energy-dependent E-I network and constraints intersection.

The gray curve describes excitatory firing rate magnitude as a function of excitatory-excitatory weights magnitude (see Eq 25), whereas the red curve conceptualizes the inverse relation between excitatory firing rates and excitatory-excitatory weights magnitudes given by metabolic constraints (see Eq 42) affecting non-silent excitatory neurons in the energy fixed point. On the blue region, the postsynaptic energy level drops below the energy level fixed point . Thus, in the blue region, experience a net depression drift (see Eq 31), whereas in the white region, the postsynaptic energy level is above . Consequently, in the white region, there is a net potentiation drift. Therefore, if the network state is in the blue region, energy-dependent plasticity rules push the network state toward the intersection between the two curves (following the green arrows). Likewise, if the network is in the white region, energy-dependent plasticity pushes the network state toward the intersection point between the two curves (following the blue arrows). Finally, in the intersection between the two curves (the system’s metabolic fixed point) the postsynaptic energy level is and, as a consequence, on average. For a detailed fixed point stability analysis, please refer to S1 Text.

https://doi.org/10.1371/journal.pcbi.1013148.g004

With an analytical understanding of how metabolic constraints affect the structure and dynamics of the network, let us now explore the impact of metabolic constraints on the network through numerical simulations.

Network activity and structure without energy constraints.

Before analyzing the network’s structure and dynamics under energy constraints on simulations, we first simulate the network without energy constraints (i.e., ). Thus, the current-rate relation from Eq 25 as well as the available energy in the neuron dictated by Eq 38 holds, but Eq 40 does not hold, because Eq 40 holds for the excitatory population only if energy constraints are affecting plasticity (i.e., ).

Simulating the network without metabolic constraints is useful as a base case for comparisons before including metabolic constraints (all the simulations presented in the Results section have the same initial conditions). Fig 5 shows the network’s dynamics and structure when no metabolic constraints are included, with panels (b) and (c) included for direct comparison to simulation with metabolic constraints. As expected, firing rates increase until saturation due to favored synaptic potentiation (). In this regard, the refractory period for each neuron in the network is . Thus, the maximum theoretical firing rate for an infinite stimulation current is (see Eq 21). Therefore, the network’s firing rate is close to the saturation state. Moreover, the mean energy level of the excitatory population (Fig 5) decreases until stabilizing at . The energy stabilization point occurs at the firing rate and excitatory-excitatory strength saturation.

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Fig 5. Network dynamic and structure without energy constraints.

Network simulation when there are no metabolic constraints in the network. Mean available energy in the excitatory population keeps dropping until neuronal firing rates are saturated. (a) shows mean energy level and mean firing rate for the excitatory population, while (b) shows final incoming synaptic strength per neuron () and mean firing rate per neuron. The mean firing rates in (b) and (c) are calculated considering the last of the simulation.

https://doi.org/10.1371/journal.pcbi.1013148.g005

The previous observation is supported by realizing that if firing rates and excitatory-excitatory weights are saturated, the energy consumption in each neuron can be approximated by:

in agreement with what is observed in Fig 5A. However, in principle, without energy constraints, each neuron’s energy level can decrease until achieving A = 0. This is the case if we, for instance, sufficiently increase the number of neurons in the network, or increase the energy consumption related to postsynaptic potentials E_syn. Eq 37 describes which variables affect energy consumption and the available energy in the neuron.

However, we do not analyze the limit in this work, as it is outside the scope of our study.

Exploring synaptic energy imbalances sensitivity .

From our previous theoretical developments, for higher synaptic energy imbalance sensitivities we expect higher energy level equilibrium values (Eq 33). To achieve closer to A_H, energy consumption needs to decrease, and postsynaptic energy consumption decreases if the presynaptic rate decreases, or if incoming synaptic strengths decrease (Eq 41). Therefore, as we increase , we expect to observe both lower firing rates and weaker excitatory-to-excitatory synaptic strengths. This trend is conceptually illustrated in Fig 4 and is numerically validated by the data presented in the first row of S3 Fig. In addition, to satisfy Eq 41 an inverse relation between and should be observable in the simulations.

Fig 6 shows the available energy per neuron and mean neuronal firing rate through the simulation, for the excitatory population with different synaptic energy imbalance sensitivity (). Consistently with Eq 33, while synaptic energy imbalance sensitivity increases, the energy level equilibrium point increases and stays close to our analytical prediction (dashed line). Also, while the energy fixed point increases, the mean neuronal firing rate decreases, which aligns with the theoretical description represented in Fig 4.

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Fig 6. Available energy and firing rate for different synaptic sensitivities to energy imbalances.

(a) Available energy per neuron for different synaptic sensitivities . (b) Mean firing rate for the excitatory population for different synaptic sensitivities . (c) Mean firing rate versus incoming synaptic strength per neuron. (d) Mean energy consumption per neuron due to presynaptic excitatory and inhibitory neurons, and the dashed lines represent the theoretical relationship described in Eq 41. In all simulation there is no neuronal sensitivity to energy imbalances (i.e., ). In (c) and (d), only the last of the simulation is considered.

https://doi.org/10.1371/journal.pcbi.1013148.g006

Fig 6C shows neuronal firing rate as a function of incoming synaptic strength, for different synaptic sensitivities values. In agreement with Eq 25, as the incoming synaptic strengths increases, the firing rate increases. Also, when increases, neuronal rates and excitatory-excitatory weights decrease, as can be observed from Fig 6C with (blue) having higher firing rates and stronger excitatory-excitatory synapses than the ones observed when (purple). The previous observation is also valid when contrasting Fig 6C for against (red). Thus, when synaptic sensitivity increases, excitatory firing rates tend to decrease.

Regarding the inverse relation between and dictated by Eq 41, Fig 6D shows the relationship between these two quantities for simulations with increasing . For a low synaptic sensitivity to energy imbalance ( (blue)), there seems to be an inverse relation, although not very clear. However, if synaptic sensitivity increases, the inverse relationship between the two variables become more evident, as shown in the case (purple). Interestingly, when the synapses are highly sensitive to energy imbalance (the case (red) in Fig 6C and Fig 6D), it is not possible for all the neurons to achieve the metabolic fixed point and some neurons remain silent (neurons with firing rate equal to zero in Fig 6C).

It is not surprising that silent neurons have higher inhibitory weights. This happens because just before the separation of the network into two subpopulations (), there is a slight decrease in available energy coexisting with an increase in the firing rate, followed by an increase in available energy joined with a reduction in the firing rate. This last increment in available energy must be accompanied by a general decrease in excitatory-excitatory weights ( implies a net negative drift in excitatory-excitatory weights, Eq 32). If there is a general decrement in excitatory-excitatory weights, then the first neurons to become silent are those with higher inhibitory incoming weights. After entering the silent state, their excitatory-excitatory weights remain constant, so the only possibility for them to stop being silent is to receive higher incoming current, but this is not possible, because the silent neurons allow a decrease in the energy consumption in the non-silent neurons, enabling the non-silent neurons to satisfy the metabolic fixed point where . As a consequence, the silence of those neurons enables the network to converge toward a stable fixed point, with one subpopulation staying in the metabolic fixed point, while another subpopulation remains in the silent fixed point. In this scenario, one group of neurons still tries to satisfy the energy constraints, and their the energy equilibrium point is close to the one predicted by Eq 40, as shown in red in Fig 6A. The second group of silent neurons has a higher available energy, but their energy consumption is not zero. This is not the only case in which there must be silent neurons to achieve a global fixed point. For instance, if the neuronal population is large, then this phenomenon also occurs (although there is no need for extreme synaptic sensitivity to energy imbalances in this case), or if a metabolic impairment is simulated, as we will show later.

Under certain conditions, the separation of excitatory populations into two subpopulations can be interpreted as a specialization. Surprisingly, under the developed framework, silent neurons emerge as a consequence of local energy constraints in the network. If these silent neurons are not present, then it is not possible to achieve a global fixed point in the network.

Including neuronal sensitivities to energy imbalances .

So far, the developed theoretical analysis of the energy-dependent network does not account explicitly for neuronal energy imbalance sensitivity . However, we know that neuronal sensitivity to energy imbalances affects neuronal firing rate. Thus, we can include the neuronal sensitivity parameter in the current-rate mapping . In particular, if the energy level in a neuron is below the homeostatic energy level (i.e., ), for the same incoming input current, the neuron’s firing rate is higher for higher , as demonstrated by Eq 5 and shown in Fig 1. Consequently, if two networks (A and B) are equal (in particular, have the same initial excitatory-excitatory weights), but the neurons in network A have higher sensitivity than neurons in network B, then network A must have higher or equal excitatory firing rates than network B. As a consequence, to achieve an energy equilibrium point, network A needs to decrease excitatory-excitatory synaptic strengths. This idea is conceptualized in Fig 4 with the gray dashed line representing the current-rate mapping when is increased. Following the previous explanation, for higher we expect the network to have weaker excitatory-excitatory synaptic strengths as well as higher firing rates.

To numerically explore the effect of modifying , we maintain synaptic sensitivity fixed and vary the neuronal sensitivity to energy imbalances . Because the synaptic sensitivity is constant for all cases in Fig 7 and the energy level fixed point is independent of the neuronal sensitivity parameter , the theory predicts that networks with different neuronal sensitivity (and all other parameters being equal) should converge towards the same energy level. The previous prediction is confirmed by observing Fig 7, where for different values, the energy fixed points are almost the same. However, as already anticipated by the theory, in the metabolic fixed point, higher values are associated with higher mean excitatory firing rates . This prediction is confirmed by observing how increasing shifts the dots to the upper-left corner in Fig 7, thus numerically demonstrating that the energy sensitivity parameter affects the population’s firing rate, but does not change the energy equilibrium point .

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Fig 7. Available energy and firing rate for different neuronal sensitivities to energy imbalances.

(a) Available energy per neuron for different synaptic sensitivities . (b) Mean firing rate for the excitatory population for different synaptic sensitivities . (c) Mean firing rate versus incoming synaptic strength per neuron. (d) Mean energy consumption per neuron due to presynaptic excitatory and inhibitory neurons, and the dashed lines represent the theoretical relationship described in Eq 41. In all simulation there is a constant synaptic sensitivity to energy imbalances (). In (c) and (d), only the last of the simulation is considered.

https://doi.org/10.1371/journal.pcbi.1013148.g007

From Fig 7D, it is clear that the relationship defined by Eq 41 is satisfied, thus respecting the inverse relationship between and .

Regarding excitatory-excitatory synaptic strength, simulations show that excitatory-excitatory weight slightly decreases as neuronal sensitivity increases. This observation is supported by realizing that when increases, the energy fixed point does not change, but the firing rates increases (Fig 7A and 7B). This is only possible because the excitatory-excitatory weights decrease, compensating for the increased firing rates and thus allowing to keep the same energy consumption. Thus, the theoretical predictions regarding the effect of increasing while keeping fixed is proved and in well agreement with numerical experiments.

Impaired metabolic production.

Regarding biology, our theoretical and simulation framework allows us to study what happens when impaired metabolic production affects neurons. This question is relevant because evidence suggests that metabolic impairments are a common cause for various neurodegenerative diseases [5–10]. Therefore, improving the understanding of how metabolic impairments affect network dynamics and structure is potentially valuable for clinical purposes.

From our theoretical developments, if the ATP rate production controlled by parameter K decreases, we expect:

1. Given that the energy level equilibrium point guaranteeing on average is independent of K (see Eq 33), if there is a mean energy level fixed point for non-silent neurons, it should stay constant as K varies.
2. By analyzing Eq 41, for non-silent neurons, if K decrease, or needs to decrease in order to satisfy the aforementioned metabolic relation. The only plastic synapses in the network are . Therefore, if the metabolic production is impaired, we expect lower firing rates as well as weaker excitatory-excitatory synaptic strengths.
3. In neurons with impaired metabolism (lower energy production rate K), slower responses to energy consumption create a delay between production and consumption, leading to oscillatory behavior in energy levels. When the initial available energy starts at the homeostatic level (A(t = 0) = A_H), but the energy fixed point is below A_H (), energy dynamics undergo a cyclic process: excitatory-excitatory weights and firing rates initially rise, increasing energy consumption and causing available energy to drop. As energy falls below the fixed point, weights and firing rates decrease until energy production surpasses consumption, raising available energy above the fixed point. This feedback loop continues until energy production and consumption align, stabilizing energy levels near the equilibrium. When neurons’ energy production is more acutely impaired, the mismatch between energy production and consumption increases, resulting in more pronounced oscillations in the system. Our analysis further shows that for sufficiently small K, the eigenvalues of the network around the fixed point become complex conjugates, generating oscillatory dynamics (see S2 Fig).

For the metabolically impaired simulations we vary K, but keep the synaptic sensitivity to energy imbalances and the neuronal sensitivity (purple traces in Fig 7 show the healthy base case for comparison, where there is no metabolic impairment in energy production). In Fig 8 it is possible to observe that for a metabolic production rate of K = 0.7, the energy fixed point is the same as the one obtained in the healthy case (Fig 7 (purple)), confirming that if there is an energy fixed point for non-silent neurons, it is independent of the ATP rate production K, in agreement with our first theoretical prediction.

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Fig 8. Available energy and firing rate for different ATP production impairment.

(a) Available energy per neuron for different ATP production rates K. (b) Mean firing rate for the excitatory population for different ATP production rates K. (c) Mean firing rate versus incoming synaptic strength per neuron. (d) Mean energy consumption per neuron due to presynaptic excitatory and inhibitory neurons, and the dashed lines represent the theoretical relationship described in Eq 41. In all simulation there is a constant neuronal and synaptic sensitivity to energy imbalances (). In (c) and (d), only the last of the simulation is considered.

https://doi.org/10.1371/journal.pcbi.1013148.g008

Moreover, comparing Fig 7D () with Fig 8D (K = 0.7), both simulations satisfy the energy relationship defined by Eq 41 (dashed lines). However, in Fig 7D (K = 1) the mean excitatory energy consumption () is much higher than in Fig 8D (K = 0.7), aligning with the second theory prediction. Additionally, contrasting Fig 7B with Fig 8B, we observe a decrease in mean firing rate for the excitatory population when ATP production rate K decreases, also supporting the second theory prediction. The decrease in excitatory population firing rate with decreasing K is also evident when comparing Fig 7C (, purple. Firing rates ranging from approximately 90 to 110 Hz) to Fig 8C (K = 0.7, blue. Firing rates ranging from approximately 50 to 90 Hz). Therefore, if energy production is impaired in any neuron, then the energy consumption of that neuron needs to decrease to achieve a fixed point.

Hence, if we keep decreasing K (more severe production impairment), energy consumption needs to drop to be consistent with the theory. When K = 0.5 (purple in Fig 8), the mean firing rate and excitatory-excitatory weights decrease compared to the K = 0.7 case. However, the network can still converge to a global fixed point if a subpopulation of neurons becomes silent. This behavior aligns with the energy constraints of the network. As the ATP production parameter K decreases, energy consumption must also decrease. To achieve this, excitatory-excitatory weights and mean excitatory firing rates decrease, resulting in a subpopulation of silent neurons to reduce energy consumption.

This phenomenon is similar to what happens when there is high synaptic sensitivity to energy imbalances ( in Fig 7), but the cause is different. With high synaptic energy imbalance sensitivity (), the energy level fixed point is close to the homeostatic energy level A_H. This requires a reduction in energy consumption to reach a global fixed point.

In the case of metabolic impairment, the energy level fixed point is not necessarily close to A_H. Here, energy consumption needs to drop because impaired energy production cannot meet high energy demands. This balance between energy consumption and production is necessary for the network to reach a fixed point. Therefore, the constraint in the metabolic impairment case is imposed by the neurons’ energy production capability, not by synaptic energy sensitivity. Despite the different causes, the outcome is similar: silent neurons in both scenarios are those with higher inhibitory synapses.

This is because, before the network divides into two subpopulations, an increase in available energy is followed by a decrease in excitatory-excitatory weights. The first neurons to become silent are those with higher inhibitory weights as they receive less incoming excitatory current.

If we continue aggravating the metabolic production impairment and the network is able to achieve a fixed point, theoretically, we expect even weaker excitatory-excitatory synaptic strengths as well as lower firing rates in the equilibrium state. Accordingly, we simulate the network with K = 0.1, emulating the case where neurons can produce energy at a rate with respect to the healthy case (i.e., K = 1). Fig 8A and 8B show that the energy production is so impaired that the network is not able to achieve an equilibrium point and stay oscillating, exploring excitatory-excitatory weights strengths which allow for achieving an equilibrium state. The problem appears to be that there are no excitatory-excitatory weights values that allow a firing rate and energy consumption which can be compensated by the impaired energy production. Specifically, an abrupt change in firing rate due to a small variation in excitatory-excitatory strengths seem to be part of the problem. Thus, small excitatory-excitatory weight variations generate energy consumption that cannot be compensated by on-demand energy production. This is coherent with the third prediction and our S1 Text (S2 Fig), highlighting an extreme case where energy production is so impaired that it is not possible to converge towards a homeostatic balance, where energy production and consumption match each other.

To understand this intuitively, it is useful to remember that neurons with impaired metabolism (small K) have slower responses to energy consumption, causing a higher delay between consumption and production. Also, excitatory-excitatory weight changes depend on the available energy in the neurons (see Eqs 32 and 2).

If we start the simulation with all neurons at the homeostatic energy level A(t = 0) = A_H, and the energy fixed point is lower than the initial energy (), then initially, weights will increase, and firing rates will rise due to the current-rate relationship. This leads to increased energy consumption, but because energy production is delayed, the available energy will tend to decrease ().

As available energy continues to drop, eventually it will be lower than the energy fixed point (). When this happens, excitatory-excitatory weights will decrease, reducing firing rates. This process will continue until energy consumption decreases enough for energy production to surpass it. When energy production surpasses consumption, the available energy in neurons starts to increase () until it matches the energy equilibrium point ().

Because energy production is always active below the fixed point (), this allows the available energy to rise above . When the available energy exceeds the equilibrium value, the cycle repeats: weights and firing rates increase, leading to higher energy consumption than the neuron can handle, causing another drop in available energy ().

This cyclical behavior stops when energy production and consumption match, maintaining available energy near the fixed point (). Achieving this equilibrium is easier when on-demand energy production can compensate for consumption in neurons. For further exploration of the impact of K on network stability, please refer to S1 Text and S2 Fig.

Accordingly to our understanding of how energy constraint affects the networks, one possible solution to solve the non-converging system due to a dramatic ATP production impairment (K = 0.1), is decreasing the synaptic sensitivity to energy imbalances . In this manner, synapses experience softer strengths updates transitions when they are close to the analytical fixed point . Thus, diminishing the oscillations due to changes in the peak energy-dependent potentiation described in Eq 20. To test this hypothesis, we simulate the network (Fig 9) with the same initial conditions and parameters as the one shown in Fig 8 with K = 0.1, but decreasing the synaptic sensitivity to energy imbalances to . The proposed solution allows for alleviating the oscillations in the system. However, the equilibrium is achieved by silencing a subpopulation of excitatory neurons. The logic behind this phenomenon is the same as the one explained in the case

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Fig 9. Energy and firing rates for a dramatic metabolic impairment with low synaptic sensitivity to energy imbalances.

Simulations are obtained with parameters. Figure shows results for the excitatory population.

https://doi.org/10.1371/journal.pcbi.1013148.g009

Fig 9 shows that non-converging networks due to a dramatic metabolic impairment can be alleviated by modifying other parameters in the network. However, there are other solutions to this problem that might alleviate the oscillations. For instance, a simple solution is to decrease the energy expenditure due to postsynaptic potentials E_syn, thus forcing a decrease in energy consumption. Another possible solution could be modifying the neuron’s time-constant . In particular, if decreases, each neuron has a slower dynamic. Thus, decreasing the slope of the current-rate relation. As a consequence, new weights and rates combinations allow to satisfy the required energy constraints, but with lower rates, thus with slower weights updates (weights update when pre and postsynaptic spikes are present. Thus, lower rates imply fewer spikes and, therefore, slower weight updates). In addition, if the slope of the current-rate mapping decreases, weights modifications should produce softer transitions in rates, thus helping to decrease the oscillations due to the sensitivity of the firing rates to weights modifications.