Can intrinsic noise induce various resonant peaks?

We theoretically describe how weak signals may be efficiently transmitted throughout more than one frequency range in noisy excitable media by kind of stochastic multiresonance. This serves us here to reinterpret recent experiments in neuroscience, and to suggest that many other systems in nature might be able to exhibit several resonances. In fact, the observed behavior happens in our (network) model as a result of competition between (1) changes in the transmitted signals as if the units were varying their activation threshold, and (2) adaptive noise realized in the model as rapid activity-dependent fluctuations of the connection intensities. These two conditions are indeed known to characterize heterogeneously networked systems of excitable units, e.g., sets of neurons and synapses in the brain. Our results may find application also in the design of detector devices.

Some systems in nature are known to process efficiently weak signals in noisy environments. A novel mechanism that explains such ability is known as stochastic resonance (SR). This is associated with the occurrence of a peak or bell-shaped dependence in the transfer of information through an excitable system as a noise source is conveniently tuned. More specifically, low or slow noise impedes detecting a relatively weak signal but, as the noise raises, the system eventually responds correlated with the signal, which shows as a peak of information transfer. The signal is again obscured at higher noise levels. This has been reported to occur in different settings, including electronic circuits, ring lasers, crayfish mechanoreceptors, ion channels, sensory neurons, hippocampus, brain stem, and cortical areas [1,2,3,4]. An intriguing issue raised is whether a given system may filter with gain in different noise regimes, which would have technological application. After the first proposal of stochastic multiresonance (SMR) [5], the existence of two or more resonant peaks has been predicted for single-mode lasers [6], surface phenomena [7], biological diversity [8] and intracellular calcium oscillations in hepatocytes [9,10], and it has also been described in somewhat more abstract settings [11,12,13,14,15,16]. Though there is no definite claim for experimental evidence of SMR yet, two recent sets of experimental data [17,18] admit such interpretation.
Here we demonstrate that a single resonant mechanism may indeed help in transmitting signals throughout different noise frequencies. More specifically, we use an explicit mathematical model -based on independent familiar empirical descriptions for both neuron units and their synaptic linksto reveal the existence of a double resonance in an experiment concerning the human tactile blink reflex [18]. Our model behavior is also consistent with recent reports on the transfer of information with different frequencies in the hippocampus [19]. On the other hand, the model here allows one to modify the separation between the two peaks in one order of magnitude or more, and it may admit generalization to show more than two resonances, which makes the "device" very versatile.
Our main result suggests looking for SMR in nature as part of a needed effort to better understand how the details in excitable systems influence transmission. Previous studies of SR and SMR in nonlinear settings most often involved a source of controlled, additive noise getting rid of correlations. The case of an intrinsic, therefore uncontrolled, noise resulting from inherent activity in the medium is even more interesting and likely to occur in nature. In a cortical region, for instance, a given neuron may receive, in addition to the (weak) signal of interest, uncorrelated, i.e., just noisy signals from other neurons at frequencies that vary in time during the normal operation of the system. Following previous efforts [20,21], we consequently investigated the possibility of having SMR associated with the fact that both the main signal and the noise transmit through dynamic connections, e.g., synapses, whose weights change with time and, therefore, constantly modulate transmission. We found in this way that short-term activity-dependent "fatigue plasticity" -such as synaptic depression and facilitation which is known to modify the neural response causing complex behavior [22,23,24,25,26]-may indeed produce SMR in a model of neural media in agreement with recent observations. The setting in this paper, which may be seen as an application of a general study within a biological context [21], intends both to serve as a simple illustration of our point and to make contact with a specific experiment. However, it is sensible to anticipate that the main outcome here may hold rather generally in excitable systems, given that these seem to share all the relevant features in our model [27,28]. Consider a networked system in which units, say neurons, receive: (i ) a weak signal from other brain areas and/or from the senses or whatever external terminals and, in addition to this, (ii ) uncorrelated, noisy signals from other units. The latter signals will be portrayed by means of action potentials (AP) -from the many presynaptic neurons to the postsynaptic neuron-whose rates follow a Poisson distribution with mean f [29]. Besides the stochasticity this implies, we shall imagine the neurons connected by dynamic, activity-dependent links. To be specific, we shall adopt the model of dynamic synapses in [30]. That is, any synaptic link, say i, is essentially stochastic which is implemented assuming it composed of an arbitrary number, M i , of functional contacts, each releasing its transmitter content with probability u when an AP signal from other units arrives. Furthermore, to implement excitability (and, more important here, kind of threshold fickleness), the contact is assumed to become inactive afterwards for a time interval, τ ; this is a random variable with exponential distribution p t (τ ) of mean τ rec at time t. Therefore, each activation event, i.e., the arrival of an AP at i at time t i generates a (postsynaptic) signal, I i (t), which evolves according to Here, J i,ℓ is the modification in the signal produced by the AP in contact ℓ after the release event, and x i,ℓ (t) = 1 when the contact is activated, which occurs with probability u [1 − p t (τ )] , and 0 otherwise. The time constant τ in is a measure of the transmission duration (of order of milliseconds for a known type of fast postsynaptic receptors). For N units, the total postsynaptic signal is We also assume, as in [30], that both the number and the strength of functional contacts that a presynaptic unit i establishes, namely, M i and J i,ℓ vary with i according to Gaussian distributions of mean and standard deviation (M, ∆ M ) and (J, ∆ J ) , respectively. To compare with a specific experiment, we assume M = 50 ± 0.1 contacts, J = 0.3 ± 0.1 mV, u = 0.5, τ in = 1 ms and τ rec = 500 ms [31]. Just for simplicity [35], we consider a weak, low-frequency sinusoidal signal S(t) ≡ d s cos(2πf s t) which is transmitted to the (postsynaptic) unit to monitor the corresponding response and the conditions in which resonance occurs. With this aim, we then compute the generated voltage, V (t), assuming a generic dynamics of the form: where the function F is to be determined. Once the links are determined, specifying F means adopting a model for each unit. A familiar choice is the integrate-and-fire (IF) model in which F is linear with V [36]. This assumption of a fixed firing threshold is a poor description for most purposes [37], however. Instead, one could assume a networked stochastic set of (postsynaptic) units -e.g., a convenient adaptation of the network model in [27,28]-but, for the shake of simplicity, we shall inspire ourselves here in the FitzHugh-Nagumo (FHN) model [38]. The excitability is then implemented assuming that the thresholds for neuron shoots constantly adapt to the input current, which is realistic for neuron media [39]. Summing up, the unit dynamics is whereS(t) = S(t) + ρI N (t) is the input, with ρ a resistance that transforms the current I N into a voltage. W (t), which stands for a (slow recovery) variable accounting for the refractory time of the unit, satisfies: In order to compare with the experiment of interest, we shall take a = 0.001, b = 3.5 ms −1 , c = 1 ms −1 , and ǫ = 1000 ms −1 which makes the (dimensionless) voltage V (t) = 1 to correspond to 100 mV and the time variable to be within the ms range. We further assume a membrane resistance ρ = 0.1 GΩ/mV and a time constant τ m = 10 ms both within the physiological range [40]. The degree of correlation between the input signal and the output V (t) is defined as Here, ν(t) is the instantaneous firing rate of the postsynaptic unit, that is, the average number of AP's generated at time t as a consequence of inputS.
(In practice, the average is over a set of different postsynaptic AP's trains generated under the same experimental conditions.) The function C (f ) that follows from this is shown as a solid line in figure 1. As said above, previous studies illustrated SR as a peak of C when one varies the level of a noise which is apropos injected in the system. In our excitable model system, however, is the synaptic current I N (t) -and not an external noise-what directly affects dynamics. Tuning the level of noise now means increasing the frequency f of the uncorrelated AP's that are responsible for the generation of I N (t). The noise embedded in the AP's trains does not directly affect the unit, and this has a strong consequence on the shape of C. That is, SMR is then a consequence of interplay between short-term (synaptic) plasticity and threshold variability associated to dynamics (2).
To be more specific, let us write the total signal as I N =Ī N ± σ I , wherē I N > 0. In the IF model (fixed threshold),Ī N tends to reduce the voltage needed for the generation of an AP, so that the excitability of the neuron increases withĪ N . In the FHN model, however, the main effect ofĪ N is to  Figure 1: The function C (as defined in the text but scaled, arbitrary units so that it measures relative variations of the relevant correlation), according to the experimental data in [18] (full squares with their error bars) and our prediction (solid line). The empty symbols are the response when the signal S (t) , instead of the sinusoidal (therefore, time correlated) one producing the solid line, consists of a train of (uncorrelated) Poissonian pulses which, as the main signal, also endure the model synaptic dynamics; the only noticeable change is that the response results more noisy in this case due to extra randomness. The dashed line corresponds to the interpretation of these data given in [18]. (The parameter values used in these plots are well within the corresponding physiological range; see [41] and the main text for details.) move the stationary solution of system (3)-(4) (and the V nullcline) towards more positive values, so that both the resting voltage value and the voltage threshold become more positive. Then, for the range ofĪ N values of interest here, the neuron excitability depends more on the fluctuation σ I than onĪ N . On the other hand, for dynamic synapses, σ I = σ I (f ) has a non-monotonic dependence on f -it first increases from 0, reaches a maximum, say f * , and then decreases with increasing f to cero again. As a result, if the level of fluctuations at f * is such that the unit is above threshold, there will be two frequency values for which -according to familiar arguments [1]fluctuations may eventually overcome the potential barrier between silent and firing states, which results in two resonant peaks.
The model here allows one to understand, even semi-quantitatively recent data by Yasuda et al. [18] showing how short-term synaptic depression causes stochastic resonance in human tactile blink reflex. These authors monitored an input-output correlation function between tactile signal and blink reflexes associated with postsynaptic responses of neurons in the caudal pontine reticular nucleus (PnC) of the brainstem [42]. In addition to the (weak) tactile signal, these neurons received uncorrelated auditory inputs that are viewed as a noise background. Yasuda et al. then concluded that, for realistic short-term depression parameters, the postsynaptic neuron acts as an error-free detector. That is, the value of the input-output correlation function is maintained at optimal high values for a wide range of background noisy rates.
A close inspection of the Yasuda et al. data from the perspective above reveals some significant discrepancies of the fit in that study at low noise rates. That is, while experiments for low noise rates show a high inputoutput correlation level (see Table I in Ref. [18]), the theory -based on the oversimplified, linear IF neuron model with fixed firing threshold-they use to interpret and compare with their data does not predict SMR but a very low level of correlation at low frequency which is not supported by data (consequently, the authors in [18] excluded their low frequency data from their analysis). This is shown in figure 1. The disagreement may be justified by accepting that, at such low rate and due to the high neuron threshold in PnC area, the auditory noise is not enough to evoke a postsynaptic response correlated to the signal. So the high level of the correlation observed can only be understood by the effect of noise coming from other brain areas. Those authors did not study this additional noise source, however, so that the question of whether other brain areas play a role here remains unanswered. On the other hand, if such a noise is relevant, its effect should be a constant noise added to the auditory noise. Therefore, it should induce a constant increment in the noise level, which cannot explain two local maxima apparently observed in the experimental correlation data ( fig.1) at noise levels around 1 and 50 Hz, respectively.
The drawing of data in the plot of figure 1, particularly those for small f undervalued in [18], requires a comment [41]. That is, one needs to use a specific relationship between the auditory noise and f. Let us assume [43] that the firing rate of neuron i in the PnC area induced by an auditory input A is f i = f 0 + αAΘ(A − A i ), where f 0 is the level of activity in absence of any input, α is a constant, Θ(x) is the step function, and A i is the minimum input needed to induce non-spontaneous activity in neuron i. The known variability of the firing thresholds in most, e.g., PnC neurons [42], suggests one to sample A i from a Gaussian distribution with mean A 0 and variance σ 2 A . It then follows that the mean firing rate (in Hz) induced in the PnC area by an auditory input A (in dB) is This, which generalizes the linear relationship used in [18] within a restricted range, transforms all levels of auditory noise (between 30 and 90 dB in the experiment of interest) into the frequency domain. For A ≫ A 0 , (6) reduces to a linear relation.
Summing up, our model system predicts two maxima, and not only one, in the transfer of information during the specific situation that we show in fig.1. This, to be interpreted as SMR phenomena, provides a priori a good description of the only sufficiently detailed observation we know, namely, it fits all the data in [18], and it is also in qualitative agreement with several predictions, as mentioned above, and with the global behavior reported in various experiments [17,19]. The minimum which is exhibited between the two peaks is to be associated to noise-induced firings that are uncorrelated with the signal. The occurrence of an extra peak at low frequency, which is also suggested by experimental data in [44], is most interesting, e.g., as a way to efficiently detect incoming signals along two well defined noise levels. This seems to occur in nature and could also be implemented in man-made devices. The number of peaks and the frequency range at which they are  That is, an integrate-and-fire (IF) model unit can just produce a single resonance, and the same is true for FitzHugh-Nagumo units and for IF units with varying activation threshold as far as the connections are frozen. Nevertheless, these two model units can produce two resonance peaks if the connections are dynamic, e.g., if they show short-time fatigue plasticity which is known to occur in many networks in nature. located can easily be controlled in the model by tuning parameter values, particularly those concerning synaptic dynamics.
Finally, we remark the model indication of two main ingredients of SMR. On one hand, the system is expected to have activity-dependent excitability. This may require short-term variations of intensity links in a networked system, which is very common in practice [27]. On the other hand, the units in our model are able to adapt activation or firing thresholds to the level of mean input. It is sensible to expect such adaptive thresholds [45,21], and they have been observed recently in actual cortical regions [46], for instance. A main conclusion is therefore that SMR should be observed rather generally in neural media and in other excitable systems. We summarize in figure 2 the conditions in which such an interesting phenomenon may occur. Incidentally, it is also worthwhile mentioning that the present work adds to previous efforts analyzing the consequences in many branches of science of the interplay between nonlinearities, signal and forces, and environmental noise [47].