Sketch of the model architecture and biological examples at different scales. (a) A receptor R transitions between an active (A) and passive (P) state along two pathways, one used for sensing (red) and affected by the environment h, and the other (blue) modified by the storage concentration, [S]. An active receptor increases the response of a readout population U (orange), which in turn stimulates the production of storage units S (green) that provide negative feedback to the receptor. (b) In tumor necrosis factor (TNF) signaling, we can identify a similar architecture. The nuclear factor NF 𢈒 κB is produced after receptor binding to TNF. NF − κB modulates the encoding of the zinc-finger protein A20, which closes the feedback loop by inhibiting the receptor complex. (c) Similarly, in olfactory sensing, odorant binding induces the activation of adenylyl cyclase (AC). AC stimulates a calcium flux, eventually producing phosphorylase calmodulin kinase II (CAMKII) which phosphorylates and deactivates AC. (d) In neural response, multiple mechanisms take place at different scales. In zebrafish larvae, visual stimulation is pro jected along the visual stream from the retina to the cortex, a coarse-grained realization of the R-U dynamics. Neural habituation emerges upon repeated stimulation, as measured by calcium fluorescence signals (dF/F0) and by the corresponding 2-dimensional PCA of the activity profiles.

Evolution of the model under a switching external field H(t). (a) The change in the average readout population Δ 〈U〉 between the first signal and after a large number of signals, as a function of the inverse temperature β and the energetic cost of storage σ. Δ 〈U〉 quantifies the habituation strength. (b) The change in the mutual information between the readout population and the external field, ΔIU,H. A region with maximal information gain corresponds to intermediate habituation strength. (c) The change in the feedback information ΔIf indicates that, close to the region of maximal information gain, the storage favors information. (d-e) In the region of maximal information gain, the average number of readout units, 〈U〉, decreases with the number of repetitions, while the average storage concentration, 〈S〉, increases. At large times, the system reaches a periodic steady state. (f-g) In the same region, the information encoded on H through the readout, IU,H, increases in time during habituation, boosting in turn the feedback information, ΔIf. (h) The internal dissipation rate due to the production of U and S, , decreases in time. Model parameters for panels (d-h) are β = 3, σ = 0.6 (in the unit measure of energy, for simplicity), and as specified in the Methods.

Optimality at the onset of habituation. (a-b) Contour plots in the (β, σ) plane of the stationary mutual information and the receptor dissipation per unit temperature, , in the presence of a constant external input. (c) For a given value of β, the system can optimize σ to the Pareto front (black line) to simultaneously minimize δQR and maximize IU,H. Each point in this space corresponds to a different strategy γ. If γ = 0, the system minimizes dissipation only, and if γ = 1 it only maximizes information. Below the front, the system exploits the available energy suboptimally, reaching lower values of information. In contrast, the region above the front is physically inaccessible. (d-e) In the presence of a dynamical input, the parameters defining the optimal front capture the region of maximal information gain and thus correspond to the onset of habituation, where 〈U〉 starts to be significantly smaller than zero. (f) Projection of ΔIU,H and Δ 〈U〉 along σ for a range of values of β ∈ [3 − 3.5]. The gray area enclosed by the dashed vertical lines indicates the location of the Pareto front for these values. ΔIU,H clearly peaks at optimality, while Δ 〈U〉 takes intermediate values.

The role of memory in shaping habituation. (a) The system response depends on the waiting time ΔTpause between two external signals. As ΔTpause increases, the storage decays and thus memory is lost (green), and consequently the habituation of the readout population decreases (yellow). (b) As a consequence, the information IU,H that the system has on the field H when the new stimulus arrives decays as well. Model parameters for this figure are β = 2.5, σ = 0.5 in the unit measure of the energy, and as specified in the Methods.

Habituation in zebrafish larvae. (a) Normalized neural activity profile in a zebrafish larva in response to the repeated presentation of visual stimulation, and comparison with the fraction of active neurons 〈Nact〉 = 〈Nact〉/N in our model with stochastic neural activation (see Methods). Stimuli are indicated with colored dots from blue to red as time increases. (b) PCA of experimental data reveals that habituation is captured mostly by the second principal direction, while features of the evoked neural response by the first one. Different colors indicate responses to different stimuli. (c) PCA of simulated neural activations. Although we cannot capture the dynamics of the evoked neural response with a switching input, the core features of habituation are correctly captured along the second principal direction. Model parameters are β = 4.5, σ = 0.15 in energy units, and as in the Methods, so that the system is tuned to the onset of habituation.

Dynamics of a system where U evolves on the same timescale of H, and implements directly a negative feedback on the receptor. In this model, 〈U〉 (in red) increases upon repeated stimulation rather than decreasing, responding to changes in 〈H〉 (in gray) as the storage of the full model. On the other hand, the probability of the receptor being active, pR(r = 1) (black), shows signs of habituation.

Effects of the external signal strength and thermal noise level on sensing. (a) At fixed and low σ = 0.1 and constant exponentially distributed signal with mean 〈H〉. As 〈H〉 decreases, the system captures less information and it needs to operate at lower temperatures to sense the signal. In particular, as the temperature decreases, IU,H becomes larger. (b) In the dynamical case, outside the optimal surface, at high β and high σ, storage is not produced and thus no negative feedback is present. The system does not display habituation, and IU, H is smaller than inside the optimal surface (gray area). (c) In the opposite regime, at low β and σ, the system is dominated by thermal noise. As a consequence, the average readout 〈U〉 is high even when the external signal is not present (〈H〉 = 〈H〉min = 0.1), and it captures only a small amount of information IU, H, which is masked by thermal activation. Other simulation parameters for this figure are 〈U〉A = 150, 〈U〉P = 0.5, β = 2/3, and = ΔE = g. For the dynamical case, Ton = Toff = 100Δt.

Trade-off between energy dissipation and information with a 3D optimization including information feedback. (a) The optimal surface in the (IU, H, ΔIf, −δQR) space with a constant external field, is obtained through a Pareto-like optimization. (b-d) The values of σ and β inside the optimal surface (gray surface) maximize both the readout information, IU, H, and the information feedback of the storage population, ΔIf, while minimizing the dissipation of the receptor, δQR. (e) At the optimal (β, σ) (gray area, shown at a fixed value of σ « 1.5) the average readout, 〈U〉, and storage, 〈S〉, are at intermediate values.

Behavior of the average readout population, 〈U〉, the average storage population, 〈S〉, the mutual information between them, IU, S, and the entropy production of the internal processes, , as a function of β and σ and in the presence of a static field. The gray area represents the Pareto-like optimal surface, while the dashed black line indicates the 2-dimensional Pareto front derived in the main text. The signal is exponentially distributed with an inverse characteristic scale λ = 0.1, so that 〈H〉 = 10. Other simulation parameters are as in Figure S7.

Dynamical optimality under a repeated external signal. (a) Schematic definition of how we study the dynamical evolution of relevant observables, by comparing the maximal response to a first signal with the one to a signal at large times. (b) Behavior of the increase in readout information, ΔIU, H, in feedback information, ΔΔIf, in average storage population, Δ〈U〉, and in entropy production, Δ. The gray area represents the Pareto-like optimal surface in the presence of a static field, while the dashed black line indicates the 2-dimensional Pareto front obtained in the same conditions. Simulations parameters are as in Figure S7. In particular, recall the signal is exponentially distributed whose characteristic scale follows a square wave, with 〈H〉max = 10, 〈H〉max = 0.1, and Ton = Toff = 100Δt.

Effect of the signal duration on habituation. (a) If the system only receives the signal for a short time (Ton = 50ΔT < Toff = 200ΔT) it does not have enough time to reach a high level of storage concentration. As a consequence, both ΔU and ΔIU, H are smaller, and thus habituation is less effective. (b) If the system receives long signals with brief pauses (Ton = 200ΔT > Toff = 50ΔT), instead, the habituation mechanism promotes information storage and thus a reduction in the readout activity. Other simulation parameters are as in Figure S10. Gray area and dashed black line represent the 3dimensional and 2dimensional optimal region where habituation emerges respectively.