# The Analysis of Mammalian Hearing Systems Supports the Hypothesis That Criticality Favors Neuronal Information Representation but Not Computation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Universality and Criticality in Physics

## 3. Application to Biology

#### 3.1. A List of Prominent Conjectures

#### 3.2. Cochlear Prototype of Neural Circuits

#### 3.3. Effects of Computation

#### 3.4. Real-World Example of EMOCS-Guided Computation

#### 3.5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Vogels, T.P.; Rajan, K.; Abbott, L.F. Neural network dynamics. Annu. Rev. Neurosci.
**2005**, 28, 357–376. [Google Scholar] [CrossRef] [PubMed] - Ringach, D.L. Spontaneous and driven cortical activity: Implications for computation. Curr. Opin. Neurobiol.
**2009**, 19, 439–444. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sussillo, D. Neural circuits as computational dynamical systems. Curr. Opin. Neurobiol.
**2014**, 25, 156–163. [Google Scholar] [CrossRef] [PubMed] - Kanders, K.; Lorimer, T.; Stoop, R. Avalanche and edge-of-chaos criticality do not necessarily co-occur in neural networks. Chaos
**2017**, 27, 047408. [Google Scholar] [CrossRef] [PubMed] - Kanders, K.; Lee, H.; Hong, N.; Nam, Y.; Stoop, R. Fingerprints of a second order critical line in developing neural networks. Commun. Phys.
**2020**, 3, 13. [Google Scholar] [CrossRef] [Green Version] - Beggs, J.M.; Plenz, D. Neuronal avalanches in neocortical circuits. J. Neurosci.
**2003**, 23, 11167–11177. [Google Scholar] [CrossRef] [Green Version] - Mazzoni, A.; Broccard, F.D.; Garcia-Perez, E.; Bonifazi, P.; Ruaro, M.E.; Torre, V. On the dynamics of the spontaneous activity in neuronal networks. PLoS ONE
**2007**, 2, e439. [Google Scholar] [CrossRef] [Green Version] - Pasquale, V.; Massobrio, P.; Bologna, L.L.; Chiappalone, M.; Martinoia, S. Self-organization and neuronal avalanches in networks of dissociated cortical neurons. Neurosciences
**2008**, 153, 1354–1369. [Google Scholar] [CrossRef] - Petermann, T.; Thiagarajana, T.C.; Lebedev, M.A.; Nicolelis, M.A.L.; Chialvo, D.R.; Plenz, D. Spontaneous cortical activity in awake monkeys composed of neuronal avalanches. Proc. Natl. Acad. Sci. USA
**2009**, 106, 15921–15926. [Google Scholar] [CrossRef] [Green Version] - Hahn, G.; Petermann, T.; Havenith, M.N.; Yu, S.; Singer, W.; Plenz, D.; Nikolić, D. Neuronal avalanches in spontaneous activity in vivo. J. Neurophysiol.
**2010**, 104, 3312–3322. [Google Scholar] [CrossRef] - Allegrini, P.; Paradisi, P.; Menicucci, D.; Gemignani, A. Fractal complexity in spontaneous EEG metastable-state transitions: New vistas on integrated neural dynamics. Front. Physiol.
**2010**, 1, 128. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Palva, J.M.; Zhigalov, A.; Hirvonen, J.; Korhonen, O.; Linkenkaer-Hansen, K.; Palva, S. Neuronal long-range temporal correlations and avalanche dynamics are correlated with behavioral scaling laws. Proc. Natl. Acad. Sci. USA
**2013**, 110, 3585–3590. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tagliazucchi, E.; Balenzuela, P.; Fraiman, D.; Chialvo, D.R. Criticality in large-scale brain FMRI dynamics unveiled by a novel point process analysis. Front. Physiol.
**2012**, 3, 15. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Stanley, H.E. Introduction to Phase Transitions and Critical Phenomena; Oxford University Press: Oxford, UK, 1987. [Google Scholar]
- Mora, T.; Bialek, W. Are biological systems poised at criticality? J. Stat. Phys.
**2011**, 144, 268–302. [Google Scholar] [CrossRef] [Green Version] - Beggs, J.M. The criticality hypothesis: How local cortical networks might optimize information processing. Philos. Trans. R. Soc. A
**2008**, 366, 329–343. [Google Scholar] [CrossRef] - Chialvo, D.R. Emergent complex neural dynamics. Nat. Phys.
**2010**, 6, 744–750. [Google Scholar] [CrossRef] [Green Version] - Hesse, J.; Gross, T. Self-organized criticality as a fundamental property of neural systems. Front. Syst. Neurosci.
**2014**, 8, 166. [Google Scholar] [CrossRef] [Green Version] - Priesemann, V.; Wibral, M.; Valderrama, M.; Pröpper, R.; Le Van Quyen, M.; Geisel, T.; Triesch, J.; Nikolic, D.; Munk, M.H. Spike avalanches in vivo suggest a driven, slightly subcritical brain state. Front. Syst. Neurosci.
**2014**, 8, 108. [Google Scholar] [CrossRef] [Green Version] - Shew, W.L.; Plenz, D. The functional benefits of criticality in the cortex. Neuroscientist
**2013**, 19, 88–100. [Google Scholar] [CrossRef] - Shew, W.L.; Yang, H.; Petermann, T.; Roy, R.; Plenz, D. Neuronal avalanches imply maximum dynamic range in cortical networks at criticality. J. Neurosci.
**2009**, 29, 15595–15600. [Google Scholar] [CrossRef] - Haldeman, C.; Beggs, J.M. Critical branching captures activity in living neural networks and maximizes the number of metastable states. Phys. Rev. Lett.
**2005**, 94, 058101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Stoop, R.; Gomez, F. Auditory power-law activation avalanches exhibit a fundamental computational ground state. Phys. Rev. Lett.
**2016**, 117, 038102. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Touboul, J.; Destexhe, A. Power-law statistics and universal scaling in the absence of criticality. Phys. Rev. E
**2017**, 95, 012413. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Martinello, M.; Hidalgo, J.; Maritan, A.; di Santo, S.; Plenz, D.; Muñoz, M.A. Neural theory and scale-free neural dynamics. Phys. Rev. X
**2017**, 7, 041071. [Google Scholar] [CrossRef] [Green Version] - Beggs, J.M.; Timme, N. Being critical of criticality in the brain. Front. Physiol.
**2012**, 3, 163. [Google Scholar] [CrossRef] [Green Version] - Gomez, F.; Saase, V.; Buchheim, N.; Stoop, R. How the ear tunes in to sounds: A physics approach. Phys. Rev. Appl.
**2014**, 1, 014003. [Google Scholar] [CrossRef] - Gomez, F.; Stoop, R. Mammalian pitch sensation shaped by the cochlear fluid. Nat. Phys.
**2014**, 10, 530–536. [Google Scholar] [CrossRef] [Green Version] - Kadanoff, L.P. Scaling laws for Ising models near Tc. Phys. Phys. Fiz.
**1966**, 6, 263–272. [Google Scholar] [CrossRef] [Green Version] - Tong, D. Statistical Field Theory; Lecture Notes; University of Cambridge: Cambridge, UK, 2017. [Google Scholar]
- Täuber, U.C. Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Zarzycki, J. Glasses and the Vitreous State; Cambridge University Press: Cambridge, UK, 1991. [Google Scholar]
- Feigenbaum, M.J. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys.
**1978**, 19, 158. [Google Scholar] [CrossRef] - Feigenbaum, M.J. Universality in Complex Discrete Dynamics; Report 1975–1976; LA-6816-PR, 98-102; Los Alamos Scientific Laboratory: Los Alamos, NM, USA, 1976. [Google Scholar]
- Stauffer, D.; Aharony, A. Introduction to Percolation Theory, 2nd ed.; CRC Press: Boca Raton, FL, USA, 1994. [Google Scholar]
- Wechsler, D.; Stoop, R. Complex structures and behavior from elementary adaptive network automata. In Emergent Complexity from Nonlinearity, in Physics, Engineering and the Life Sciences; Springer: Cham, Switzerland, 2017; Volume 191, pp. 105–126. [Google Scholar]
- Amaral, L.A.N.; Scala, A.; Barthélémy, M.; Stanley, H.E. Classes of small-world networks. Proc. Natl. Acad. Sci. USA
**2000**, 97, 11149. [Google Scholar] [CrossRef] [Green Version] - Mossa, S.; Barthélémy, M.; Stanley, H.E.; Amaral, L.A.N. Truncation of power law behavior in ‘scale-free’ network models due to information filtering. Phys. Rev. Lett.
**2002**, 88, 138701. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dorogovtsev, S.N.; Mendes, J.F.F. Language as an evolving word web. Proc. R. Soc. Lond. B
**2001**, 268, 2603. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Assenza, S.; Gutiérrez, R.; Gómez-Gardeñes, J.; Latora, V.; Boccaletti, S. Emergence of structural patterns out of synchronization in networks with competitive interactions. Sci. Rep.
**2011**, 1, 99. [Google Scholar] [CrossRef] [PubMed] - Eurich, C.W.; Herrmann, J.M.; Ernst, U.A. Finite-size effects of avalanche dynamics. Phys. Rev. E
**2002**, 66, 066137. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Levina, A.; Herrmann, J.M.; Geisel, T. Dynamical synapses causing self-organized criticality in neural networks. Nat. Phys.
**2007**, 3, 857. [Google Scholar] [CrossRef] - de Arcangelis, L.; Lombardi, F.; Herrmann, H.J. Criticality in the brain. J. Stat. Mech.
**2014**, 3, P03026. [Google Scholar] [CrossRef] - Lorimer, T.; Gomez, F.; Stoop, R. Two universal physical principles shape the power-law statistics of real-world networks. Sci. Rep.
**2015**, 5, 12353. [Google Scholar] [CrossRef] [Green Version] - Stoop, R.; Stoop, N.; Bunimovich, L.A. Complexity of Dynamics as Variability of Predictability. J. Stat. Phys.
**2004**, 114, 1127–1137. [Google Scholar] [CrossRef] - van der Waals, J.D. Over de Continuiteit van den gas—En Vloeistoftoestand; Sijthoff: Leiden, The Netherlands, 1873. [Google Scholar]
- Held, J.; Lorimer, T.; Pomati, F.; Stoop, R.; Albert, C. Second-order phase transition in phytoplankton trait dynamics. Chaos
**2020**, 30, 053109. [Google Scholar] [CrossRef] - Kauffman, S.A. The Origins of Order; Oxford University Press: Oxford, UK, 1993. [Google Scholar]
- Kauffman, S.A. Metabolic stability and epigenesis in randomly constructed genetic nets. J.Theor. Biol.
**1969**, 22, 437–467. [Google Scholar] [CrossRef] - Bak, P.; Tang, C.; Wiesenfeld, K. Self-organized criticality: An explanation of 1/f noise. Phys. Rev. Lett.
**1987**, 59, 381–384. [Google Scholar] [CrossRef] - Olami, Z.; Feder, H.J.S.; Christensen, K. Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes. Phys. Rev. Lett.
**1992**, 68, 1244–1247. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Drossel, B.; Schwabl, F. Self-organized critical forest-fire model. Phys. Rev. Lett.
**1992**, 69, 1629–1632. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bak, P.; Sneppen, K. Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Lett.
**1993**, 71, 4083–4086. [Google Scholar] [CrossRef] [PubMed] - Langton, C. Studying artificial life with cellular automata. Physica D
**1986**, 22, 120–149. [Google Scholar] [CrossRef] [Green Version] - Packard, N. Adaptation Toward the Edge of Chaos. In Dynamic Patterns in Complex Systems; World Scientific: Singapore, 1988. [Google Scholar]
- Crutchfield, J.P.; Young, K. Computation at the Onset of Chaos. In Entropy, Complexity, and the Physics of Information; Zurek, W., Ed.; SFI Studies in the Sciences of Complexity, VIII; Addison-Wesley: Reading, MA, USA, 1990; pp. 223–269. [Google Scholar]
- Bak, P.; Tang, C. Earthquakes as a self-organized critical phenomenon. J. Geophys. Res.
**1989**, 94, 635–637. [Google Scholar] [CrossRef] [Green Version] - Harris, T.E. The Theory of Branching Processes; Dover Publications: New York, NY, USA, 1989. [Google Scholar]
- Zapperi, S.; Lauritsen, K.B.; Stanley, H.E. Self-organized branching processes: Mean-field theory for avalanches. Phys. Rev. Lett.
**1995**, 75, 4071–4074. [Google Scholar] [CrossRef] [Green Version] - Tetzlaff, C.; Okujeni, S.; Egert, U.; Wörgötter, F.; Butz, M. Self-organized criticality in developing neuronal networks. PLoS Comput. Bio.
**2010**, 6, e1001013. [Google Scholar] [CrossRef] [Green Version] - Shew, W.L.; Clawson, W.P.; Pobst, J.; Karimipanah, Y.; Wright, N.C.; Wessel, R. Adaptation to sensory input tunes visual cortex to criticality. Nat. Phys.
**2015**, 11, 659–664. [Google Scholar] [CrossRef] [Green Version] - Ribeiro, T.L.; Ribeiro, S.; Belchior, H.; Caixeta, F.; Copelli, M. Undersampled critical branching processes on small-world and random networks fail to reproduce the statistics of spike avalanches. PLoS ONE
**2014**, 9, e94992. [Google Scholar] - Yaghoubi, M.; de Graaf, T.; Orlandi, J.G.; Girotto, F.; Colicos, M.A.; Davidsen, J. Neuronal avalanche dynamics indicates different universality classes in neuronal cultures. Sci. Rep.
**2018**, 8, 3417. [Google Scholar] [CrossRef] [PubMed] - Sethna, J.P.; Dahmen, K.A.; Myers, C.R. Crackling noise. Nature
**2001**, 410, 242–250. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sethna, J.P. Statistical Mechanics: Entropy, Order Parameters and Complexity; Oxford University Press: Oxford, UK, 2006. [Google Scholar]
- Stoop, R.; Stoop, N. Natural computation measured as a reduction of complexity. Chaos
**2004**, 14, 675–679. [Google Scholar] [CrossRef] [PubMed] - Gomez, F.; Lorimer, T.; Stoop, R. Signal-coupled subthreshold Hopf-type systems show a sharpened collective response. Phys. Rev. Lett.
**2016**, 116, 108101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kern, A.; Stoop, R. Essential role of couplings between hearing nonlinearities. Phys. Rev. Lett.
**2003**, 91, 128101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Martin, P.; Bozovic, D.; Choe, Y.; Hudspeth, A.J. Spontaneous oscillation by hair bundles of the bullfrog’s sacculus. J. Neurosci.
**2003**, 23, 4533–4548. [Google Scholar] [CrossRef] [Green Version] - Martignoli, S.; Gomez, F.; Stoop, R. Pitch sensation involves stochastic resonance. Sci. Rep.
**2013**, 3, 2676. [Google Scholar] [CrossRef] [Green Version] - Mountain, D.C.; Hubbard, A.E. Computational analysis of hair cell and auditory nerve processes. In Auditory Computation; Hawkins, H.L., McMullen, T.A., Popper, A.N., Fay, R.R., Eds.; Springer: New York, NY, USA, 1996; pp. 121–156. [Google Scholar]
- Lopez-Poveda, E.A.; Eustaquio-Martín, A. A biophysical model of the inner hair cell: The contribution of potassium current to peripheral compression. J Assoc. Res. Otolaryngol.
**2006**, 7, 218–235. [Google Scholar] [CrossRef] [Green Version] - Meddis, R.; Popper, A.N.; Lopez-Poveda, E.; Fay, R.R. Computational Models of the Auditory System; Springer: New-York, NY, USA, 2010. [Google Scholar]
- Wiesenfeld, K.; McNamara, B. Period-doubling systems as small-signal amplifiers. Phys. Rev. Lett.
**1985**, 55, 13–16. [Google Scholar] [CrossRef] - Wiesenfeld, K.; McNamara, B. Small-signal amplification in bifurcating dynamical systems. Phys. Rev. A
**1986**, 33, 629–642. [Google Scholar] [CrossRef] - Eguíluz, V.M.; Ospeck, M.; Choe, Y.; Hudspeth, A.J.; Magnasco, M.O. Essential nonlinearities in hearing. Phys. Rev. Lett.
**2000**, 84, 5232–5235. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Camalet, S.; Duke, T.; Julicher, F.; Prost, J. Auditory sensitivity provided by self-tuned critical oscillations of hair cells. Proc. Natl. Acad. Sci. USA
**2000**, 97, 3183–3188. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Guinan, J.J. Olivocochlear efferents: Anatomy, physiology, function, and the measurement of efferent effects in humans. Ear Hear.
**2006**, 27, 589–607. [Google Scholar] [CrossRef] [PubMed] - Cooper, N.P.; Guinan, J.J. Efferent-mediated control of basilar membrane motion. J. Physiol.
**2006**, 576, 49–54. [Google Scholar] [CrossRef] [PubMed] - Ruggero, M.A.; Rich, N.C.; Recio, A.; Narayan, S.S.; Robles, L. Basilar- membrane responses to tones at the base of the chinchilla cochlea. J. Acoust. Soc. Am.
**1997**, 101, 2151–2163. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kern, A.; Stoop, R. Principles and typical computational limitations of sparse speaker separation based on deterministic speech features. Neural Comput.
**2011**, 23, 2358–2389. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Russell, I.J.; Murugasu, E. Medial efferent inhibition suppresses basilar membrane responses to near characteristic frequency tones of moderate to high intensities. J. Acoust. Soc. Am.
**1997**, 102, 1734–1738. [Google Scholar] [CrossRef]

**Figure 1.**Cochlea’s embedding into mammalian neural circuitry [78,79]. Upper panel: Brainstem section (CN: cochlear nucleus, EMOC: medial olivocochlear efferents, SOC: superior olivary complex, FV: fourth ventricle). Lower panel: Section (dashed green line) of organ of Corti. EMOC stimulations tune the cochlea’s sensitivity (see section ‘Effects of computation’).

**Figure 2.**Hopf small signal amplifiers describe the effect of outer hair cells (Hopf frequency ${\omega}_{ch}=10$, dB output (input) in terms of $10{log}_{10}{\left|a\right|}^{2}$ ($10{log}_{10}{\left|{F}_{0}\right|}^{2}$)). (

**a**) Output curves as a function of the input frequency $\omega $ for different values of $\mid {F}_{0}\mid $ between ${10}^{-4}$ and 1 and $\mu =-0.01$. (

**b**) Output at resonance ($\omega ={\omega}_{ch}$) for input ${F}_{0}$ using different $\mu $-values. Dotted: off-resonance curve ($\omega =9.97$, $\mu =-0.01$), for comparison. Note that the amplifier is subcritically tuned, not critically, and isolated from the cochlear environment.

**Figure 3.**‘Close-to-biology small-signal’ amplifier implementation, including subcritical tuning and influence of cochlear fluid (endolymph). Hopf cochlea response to pure tones. (

**a**) Response in dB, (

**b**) gain in dB; a difference of 33 dB in peak gain for two input levels differing by 70 dB corresponds to observations in chinchilla (32.5 dB, or slightly higher, between 20 and 90 dB SPL curves [80]). (

**c**) Tuning curves for fixed output levels, (

**d**) Phase for different input levels relative to −30 dB. (Cochlea discretization covering 14.08–0.44 kHz (20 sections); output at section 5 (CF = 6.79 kHz) for (

**a**–

**c**) and section 3 (CF = 9.78 kHz) for (

**d**). Small numbers denote input levels in dB).

**Figure 4.**Traveling wave along the cochlea, from the stapes side (hair cells responsive towards high-frequency stimulations) towards the apex (hair cells responding to low frequencies). Input frequencies are 8, 3, 1 kHz in (

**a**–

**c**), respectively. Upper panel: Extrapolated continuous excitation, lower panel: activation patterns on the artificial cochlea.

**Figure 5.**(

**a**) Biological vs. (

**b**) artificial implementation of the hearing–listening circuit. Listening is a dedicated activity that represents a particular computational effort, involving ‘EMOCS’ (efferent medial olivocochlear stimulations), cf. Ref. [81].

**Figure 6.**From two pure input tones of ${f}_{1}=2200$ and ${f}_{2}=2400$ Hz, each at $-74$ dB sound level (

**a**,

**b**), additional ‘combination’ tones are generated (cf. (

**c**–

**f**)), due to the nonlinearities of the amplifiers at differences of the input frequencies, of essentially exponentially decaying amplitudes.

**Figure 7.**Activation networks from (

**a**) two pure tones (3/8, 1/2 kHz), (

**b**) two complex tones (2, 3.35 kHz, 5 harmonics each).

**Figure 8.**Cochlea activity A using uniformly chosen amplifier parameter values $\mu $ (range 0.1–0.3) and N = 10,000 pairs of random base frequency complex tones, from (

**a**) sound levels random from the interval $(-80,-40)$ dB (rms) per tone, (

**b**) from $-60$ dB fixed sound levels (dashed power law guidelines, exponents $\beta $ from maximum likelihood estimation: (

**a**) $\beta =0.6,0.44,0.3,0.2,0.13$, (

**b**) $\beta =0.64,0.43,0.29,0.2,0.12$ (bottom to top lines)). Insets: Three-tone results.

**Figure 9.**EMOCS effects: (

**a**) Gain isointensity curves at section 5 (${f}_{ch}=1.42$ kHz) without (solid lines) and with (dashed lines) EMOC input. From flat tuning ($\mu =-0.1$ for all sections), EMOCS is implemented by shifting ${\mu}_{5}$ to $-1.0$ ($-80$ and $-100$ dB lines collapse). (

**b**) Corresponding phase shift at section 5 (phase delays for frequencies below CF, phase leads above CF). (

**c**) Comparison to animal data: 16 and 19 kHz pure-tone EMOCS (left and right, respectively) implemented by a shift from a flat tuned cochlea from ${\mu}_{2}=-0.05$ to ${\mu}_{2}=-0.5$ lead to BM level shifts at section 2 (${f}_{ch}=16.99$ kHz) from open circles to full circles. Insets: Corresponding experimental animal data [82].

**Figure 10.**Effects by EMOCS: (

**a**) Activation distributions of the experiment of Figure 8 after detuning Hopf sections $19,20,21$ from ${\mu}_{19,20,21}=-0.2$ (dashed), jointly to ${\mu}_{19,20,21}=-0.3,-0.5,-1,-2$, respectively, at input level $-60$ dB. (

**b**) Detuning of sections 11,12 from ${\mu}_{11,12}=-0.25$ jointly to ${\mu}_{11,12}=-1.0$, for the input of two complex tones at $-70$ dB rms each, with ${f}_{0}=1331,2120$ Hz and five harmonics.

**Figure 11.**(

**a**) Network size distributions (40,000 stimulations with two complex tones, random amplitudes) for flat tuning (blue), and after detuning two frequency bands (sections 15–16, 19–21) from $\mu =-0.25$ to $\mu =-1.0$ (red crosses), and $\mu =-2.0$ (red stars). (

**b**) In the thermodynamic formalism, the observability O of an invariant measure $\epsilon $ decays with time t as $O(\epsilon ,t)\sim {e}^{t(\epsilon -S(\epsilon \left)\right)}$ (red and blue arrows). States represented by entropy values on the diagonal $\epsilon =S\left(\epsilon \right)$ do not experience any temporal decay. Blue: Entropy function $S\left(\epsilon \right)$ of systems with power law distribution characteristics. Red: Entropy function associated with non-power-law distributions, as the result of focusing on a particular measure (green circle), where horizontal green arrows symbolize the effect by EMOCS (adapted from Ref. [23]).

**Figure 12.**Sounds of a cornett and a flute at the same fundamental frequency ${f}_{0}$ (

**left**), superimposed (

**right**), static case.

**Figure 13.**Separation of sounds, dynamic case, where the target instrument changes the height of the generated tone: (

**a**) Tuning patterns, dynamical case. Colors indicate the Hopf parameter values of the sections. Left: Cornett vs. flute (disturber). Right: Flute vs. cornett (disturber). (

**b**) TE for the two target signals of (

**a**). Black: flat tuning. Red: $\mu $-tuning. Full: cornett target, dashed: flute target. (

**c**) NSACF, NACF for the two target signals of (

**a**,

**b**) at a chosen target ground frequency. Black: flat tuning. Red: $\mu $-tuning. Blue: target signal. Targets at 392 Hz, disturbers at 2216 Hz. From Ref. [27].

**Figure 14.**TE improvement by $\mu $-tuning, static case. (

**a**) Frequency spectrum at section 8 ($CF=1964$ Hz). Blue: Flat tuning (−80 dB, target cornett ${f}_{0}=392$ Hz, disturber flute $f=2216$ Hz). Cross-combination tones (CT, two explicitly labeled) between the flute’s fundamental f and higher harmonics of the cornett are clearly visible. Red: Optimized tuning. f (flute) and cross-combination frequencies are suppressed, leaving a harmonic series of the target (small arrows). (

**b**) Averaged TE over 13 different fundamental target frequencies (steps of 1 semitone) demonstrates input amplitude independence. Blue lines: flat tuning. Red lines: optimized $\mu $-tuning. Left panel: (full lines) target sound cornett (277 to 554 Hz), disturbing sound flute (at 277 Hz); (dashed lines) same target but flute at 2216 Hz. Right panel: same experiment with target and disturber interchanged. TE improvements: arrows in (

**b**). From Ref. [27].

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Stoop, R.; Gomez, F.
The Analysis of Mammalian Hearing Systems Supports the Hypothesis That Criticality Favors Neuronal Information Representation but Not Computation. *Entropy* **2022**, *24*, 540.
https://doi.org/10.3390/e24040540

**AMA Style**

Stoop R, Gomez F.
The Analysis of Mammalian Hearing Systems Supports the Hypothesis That Criticality Favors Neuronal Information Representation but Not Computation. *Entropy*. 2022; 24(4):540.
https://doi.org/10.3390/e24040540

**Chicago/Turabian Style**

Stoop, Ruedi, and Florian Gomez.
2022. "The Analysis of Mammalian Hearing Systems Supports the Hypothesis That Criticality Favors Neuronal Information Representation but Not Computation" *Entropy* 24, no. 4: 540.
https://doi.org/10.3390/e24040540