Abstract
This chapter presents a hypothesis that the animal’s brain is acting analogously to a heat engine when it actively modulates incoming sensory information to achieve enhanced perceptual capacity. To articulate this hypothesis, we describe stimulus-evoked activity of a neural population based on the maximum entropy principle with constraints on two types of overlapping activities, one that is controlled by stimulus conditions and the other, termed internal activity, that is regulated internally in an organism. We demonstrate that modulation of the internal activity realizes gain control of stimulus response, and controls stimulus information. The model’s statistical structure common to thermodynamics allows us to construct the first law for neural dynamics, equation of state, and fluctuation-response relation. A cycle of neural dynamics is then introduced to model information processing by the neurons during which the stimulus information is dynamically enhanced by the internal gain-modulation mechanism. Based on the conservation law of entropy, we demonstrate that the cycle generates entropy ascribed to the stimulus-related activity using entropy supplied by the internal mechanism, analogously to a heat engine that produces work from heat. We provide an efficient cycle that achieves the highest entropic efficiency to retain the stimulus information. The theory allows us to quantify efficiency of the internal computation and its theoretical limit, which can be used to test the hypothesis.
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Notes
- 1.
The activity rates of Neuron 4, 5 do not depend on α because b 0 does not contain interactions that relate Neuron 1–3 with Neuron 4, 5. If there are non-zero interactions between any pair from Neuron 1–3 and Neuron 4, 5 in b 0, the activity rates of Neuron 4, 5 increase with the increased rates of Neuron 1–3.
- 2.
We obtain dU = TdS − fdX, using β ≡ 1/T and α ≡ βf in Eq. (11.11). In this form, the expectation parameter U is a function of (S, X). According to the conventions of thermodynamics, we may call U internal energy, T temperature of the system, and f force applied to neurons by a stimulus. It is possible to describe the evoked activity of a neural population using these standard terms of thermodynamics. However, this introduces the concepts of work and heat, which may not be relevant quantities for neurons to exchange with environment.
- 3.
Importantly, − ψ is a logarithm of the simultaneous silence probability predicted by the model, Eq. (11.4). The observed probability of the simultaneous silence could be different from the prediction if the model is inaccurate. For example, an Ising model may be inaccurate, and it was shown that neural higher-order interactions may significantly contribute to increasing the silence probability (Ohiorhenuan et al. 2010; Shimazaki et al. 2015).
- 4.
Under the assumption that rates of synchronous spike events scale with \(\mathcal {O}(\varDelta ^k)\), where Δ is a bin size of discretization and k is the number of synchronous neurons. It was proved (Kass et al. 2011) that it is possible to construct a continuous-time limit (Δ → 0) of the maximum entropy model that takes the synchronous events into account. Here we follow their result to consider the continuous-time representation.
- 5.
When α and β are both dependent on the stimulus, the Fisher information about s is given as
$$\displaystyle \begin{aligned} J(s) = \frac{ \partial \boldsymbol{\theta}(s)^\top } {\partial s} \mathbf{J} \frac{ \partial \boldsymbol{\theta}(s) } {\partial s}, \end{aligned} $$(11.19)Fig. 11.2 
The delayed gain-modulation by internal activity. The parameters of the maximum entropy model (N = 5) follow those in Fig. 11.1. (a) An illustration of delayed gain-modulation described in Eqs. (11.17) and (11.18). The stimulus increases the stimulus component α that activates Neuron 1, 2, and 3. Subsequently, the internal component β is increased, which increases the background activity of all 5 neurons. We assume a slower time constant for the gain-modulation than the stimulus activation (τ β = 0.1 and τ α = 0.05). (b) Top: Dynamics of the stimulus and internal components (solid lines, γ = 0.5). The internal component β without the delayed gain-modulation (γ = 0) is shown by a dashed black line. Middle: Activity rates [a.u.] of Neuron 1–3 with (solid red) and without (dashed black) the delayed gain-modulation. Bottom: The Fisher information about stimulus component α (Eq. (11.16)). (c) The X-α (left) and U-β (right) phase diagrams. A red solid cycle represents dynamics when the delayed gain-modulation is applied (γ = 0.5). The dashed line is a trajectory when the delayed gain-modulation is not applied to the population (γ = 0). (d) Left: The U-β phase diagrams of neural dynamics with different combinations of τ β and γ that achieve the same level of the maximum modulation (the minimum value of β = 0.9). Right: The Fisher information about the stimulus component α for different cycles. The color code is the same as in the left panel. The inset shows the Fisher information about the stimulus intensity s (Eq. (11.19))
where θ(s) ≡ [−β, α]⊤ and J is a Fisher information matrix given by Eq. (11.24), which will be discussed in the later section. We computed Eq. (11.19) using analytical solutions of the dynamical equations given as \(\alpha (t) = \frac {s t}{\tau _{\alpha }} e^{-t/ \tau _{\alpha }}\) and \(\beta (t) = 1 - \frac {s \gamma }{\tau _{\beta }-\tau _{\alpha }} \left \{ \frac {\tau _{\alpha } \tau _{\beta }}{\tau _{\beta }-\tau _{\alpha }} ( e^{-t/ \tau _{\beta }}-e^{-t/\tau _{\alpha }} ) - t e^{- t/\tau _{\alpha }} \right \}\).
- 6.
Here we use entropy synonymously with heat in thermodynamics to facilitate the comparison with a heat engine. However this is not an accurate description because the entropy is a state variable.
- 7.
This is synonymous with the statement that the first law prohibits a perpetual motion machine of the first kind, a machine that can work indefinitely without receiving heat.
- 8.
Let us consider the efficiency η achieved by an arbitrary cycle \(\mathcal {C}\) during which the internal component β satisfies β L ≤ β ≤ β H . Let the minimum and maximum internal activity in the cycle be U min and U max. We decompose \(\mathcal {C}\) into the path \(\mathcal {C}_1\) from U min to U max and the path \(\mathcal {C}_2\) from U max to U min during which the internal component is given as β 1(U) and β 2(U), respectively. Because the cycle acts as an engine, we expect β 1(U) > β 2(U). The entropy changes produced by the internal activity during the path C i (i = 1, 2) is computed as \(\varDelta S^{\mathrm {int}}_{\mathcal {C}_1} = \int _{U_{\mathrm {min}}}^{U_{\mathrm {max}}} \beta _1(U) \, dU \leq \beta _H \int _{U_{\mathrm {min}}}^{U_{\mathrm {max}}} \, dU = \beta _H (U_{\mathrm {max}}-U_{\mathrm {min}})\) and \(| \varDelta S^{\mathrm {int}}_{\mathcal {C}_2} |= |\int _{U_{\mathrm {max}}}^{U_{\mathrm {min}}} \beta _2(U) \, dU| \geq |\beta _L \int _{U_{\mathrm {max}}}^{U_{\mathrm {min}}} \, dU| = \beta _L (U_{\mathrm {max}}-U_{\mathrm {min}})\). Hence we obtain \(| \varDelta S^{\mathrm {int}}_{\mathcal {C}_2} | / \varDelta S^{\mathrm {int}}_{\mathcal {C}_1} \geq \beta _L / \beta _H \), or η ≤ η e .
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Acknowledgements
This chapter is an extended edition of the manuscript submitted to the arXiv (Shimazaki H., Neurons as an information-theoretic engine. arXiv:1512.07855, 2015). The author thanks J. Gaudreault, C. Donner, D. Hirashima, S. Koyama, S. Amari, and S. Ito for critically reading the original manuscript.
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Shimazaki, H. (2018). Neural Engine Hypothesis. In: Chen, Z., Sarma, S.V. (eds) Dynamic Neuroscience. Springer, Cham. https://doi.org/10.1007/978-3-319-71976-4_11
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