Living organisms respond and react to changes in their environment. They do so by decoding the information contained in these changes. This process occurs at all levels, including single cells. Cell signaling malfunction can result in pathologies. Understanding this information processing is thus important from basic and applied viewpoints. Changes in the environment are usually reflected in changes in the concentration of substances that bind to receptors on the plasma membrane. Changes in this binding lead to changes in the cell and, through a signaling cascade, to the response. The traditional view of this process assumes that step-like changes in the ligand concentration produce intracellular concentration changes that increase with the initial step size. This is called amplitude modulation encoding. Step-like changes in external effectors, however, can induce oscillations in some intermediaries [1–4]. This is called frequency modulation encoding. This mechanism is quite prevalent in the case of Ca²⁺ signals [5, 6] where oscillations in the intracellular Ca²⁺ concentration are known to regulate enzyme activity [7, 8] and increase gene expression efficiency [9]. Some pathways also generate responses using amplitude [10] and frequency encoding [11] redundantly.

Intracellular Ca²⁺ oscillations in non-excitable cells usually involve Ca²⁺ release from the endoplasmic reticulum (ER) through inositol trisphosphate (IP₃) receptors (IP₃Rs) [12]. IP₃R-Ca²⁺ channels need to bind IP₃ and Ca²⁺ on their cytosolic side to become open. This implies that the opening of neighboring IP₃Rs becomes coupled via the released Ca²⁺ in what is known as Calcium Induced Calcium Release [13]. Most often IP₃Rs are organized in clusters that are ~1.5−2μm apart. Ca²⁺ signals then range from blips and puffs to waves depending on whether Ca²⁺ is released from one or many clusters [14–16]. Ca²⁺-mediated inter-cluster coupling is fundamental for the propagation of waves as reflected by the changes that slow Ca²⁺ buffers induce on the signals [17–19]. The transition from local to global signals is like a percolation process [20] in which the accumulated free cytosolic Ca²⁺ reduces the percolation threshold [21]. Thus, the cytosol acts as an excitable medium whose excitability is modulated by Ca²⁺ and where Ca²⁺ pulses or spikes are the result of wave nucleation [22].

In the sustained presence of external effectors, such as hormones or neurotransmitters, cells can exhibit sequences of intracellular Ca²⁺ pulses. There is strong evidence that, most often, the interpulse times have a large random component [22–25]. Randomness is a consequence of the highly stochastic and spatially local Ca²⁺ release events (puffs) that eventually lead to a propagating Ca²⁺ wave or pulse [26]. Waves are then followed by an inhibitory process that acts globally in space [22, 24]. In order to account for these two processes, the model introduced in Skupin and Falcke [22] combines the stochastic “firing” of the pulses with a deterministic description of the inhibition. In this paper we use the model of Skupin and Falcke [22] together with more recent observations that show an exponential dependence between the mean interpulse time and the external effector concentration, C [25], to study the mutual information between C and some properties of the pulse sequence. We derive analytic results in two opposite limits that depend on the ratio between two timescales: the firing rate in the absence of inhibition, λ, and the rate of recovery from inhibition, ρ. For ρ/λ ≫ 1, pulse occurrence is limited by λ and the process is Poisson. For ρ/λ ≪ 1, pulse occurrence is limited by recovery from inhibition. We call this the Global Negative Feedback (GNF) limit. Fixing the value of the mean interpulse time, T, and considering the least informative distribution for C (uniform between 0 and a maximum value) we determine that the mutual information is largest in the GNF limit. The difference with respect to the Poisson limit is of, at most, ~1bit. This implies that the precision with which C can be inferred is at most twice as large in one limit with respect to the other. If we fix the firing rate, λ, instead we obtain the same mutual information between the interpulse time and C in the Poisson and the GNF limits. We discuss later the implications of these results. The co-existence of stochastic firing and deterministic recovery from inhibition is common to many systems, particularly, noise-driven excitable ones [27]. These systems, a paradigmatic example of which is neurons, appear in various settings. Thus, we think that our results are applicable beyond the case of intracellular Ca²⁺ pulses.

Here we compare the results on mutual information obtained for the model of section 2.1 in the Poisson (x = λ/ρ ≪ 1) and the GNF (x ≫ 1) limits.

Intracellular information is encoded in the oscillation frequency of messengers in many instances. This is the case of Ca²⁺, whose cytosolic concentration responds with sequences of pulses to the presence of constant concentrations of external effectors such as hormones or neurotransmitters. The statistical properties of the interpulse times have been studied theoretically and experimentally finding that they usually have a stochastic component, t, whose mean, T, and standard deviation, σ, are linearly related (Equation 5) [22–25]. The simple model [22] that includes the stochastic “firing” of the pulses with rate, λ, in the absence of inhibition, followed by immediate inhibition and subsequent recovery with rate, ρ [Equation 1 captured most features of the process. Here we combined this model with the observed dependence between T and the effector concentration, C, to compare the mutual information between C and the interpulse time, I(C, t), or the number of pulses, I(C, N), in the GNF (λ/ρ ≫ 1) and Poisson (λ/ρ ≪ 1) limits. We obtained that I(C, t) and I(C, N) were larger in the GNF limit and that the difference was at most ~1bit (it was < 0.5 for the values, BC_M ~ 1.6, explored experimentally in [25]]. This ~1bit difference means that the precision with which the effector concentration is inferred can at most double as the recovery rate is reduced while the mean, T, is kept fixed. In the case of I(N, C) the ~1bit difference is independent of p_C(C) (Equation 13) and determined by the slope, k, of the (linear) relationship between σ and T. We derived k analytically in both limits. The simulations of Skupin and Falcke [22] and Thurley and Falcke [24] show that this slope decreases monotonically when λ/ρ is increased. Thus, the information we obtain in the GNF limit should be the largest. In the GNF limit inhibition recovery is the rate-limiting process of pulse occurrence. Following the assumption that inhibition occurs globally in space, the model describes inhibition recovery deterministically. It is then reasonable that the standard deviation of the interpulse time decrease as the rate-limiting process goes from being purely stochastic (limited by λ) to being more deterministic (limited by ρ). Having a more predictable interpulse time for a given, C, should result also in a more faithful transmission of the information as we have found. As discussed in Thurley and Falcke [24] the existence of a global inhibition process is fundamental for the functional robustness of the signaling. Namely, it is the reason why Equation (5) holds with the same k for individual cells of the same type. Equation (9) also implies that, if ρ is varied for fixed λ (not T), there is no gain in the mutual information between t and C. This means that by solely reducing the recovery rate (e.g., by changing the level of expression of proteins involved in the inhibition) cells would increase the interpulse time and its variance without changing the information they could draw from the environment.

SP designed research and wrote the paper. AG performed analytic calculations and numerical simulations. AG and SP wrote Supplementary Material file.