Computation in biological systems is fundamentally energy-constrained, yet standard theories of computation treat energy as freely available. Here, we argue that variational free energy minimization under a Poisson assumption offers a principled path toward an energy-aware theory of computation. Our key observation is that the Kullback-Leibler (KL) divergence term in the Poisson free energy objective becomes proportional to the prior firing rates of model neurons, yielding an emergent metabolic cost term that penalizes high baseline activity. This structure couples an abstract information-theoretic quantity -- the *coding rate* -- to a concrete biophysical variable -- the *firing rate* -- which enables a trade-off between coding fidelity and energy expenditure. Such a coupling arises naturally in the Poisson variational autoencoder (P-VAE) -- a brain-inspired generative model that encodes inputs as discrete spike counts and recovers a spiking form of *sparse coding* as a special case -- but is absent from standard Gaussian VAEs. To demonstrate that this metabolic cost structure is unique to the Poisson formulation, we compare the P-VAE against Grelu-VAE, a Gaussian VAE with ReLU rectification applied to latent samples, which controls for the non-negativity constraint. Across a systematic sweep of the KL term weighting coefficient $β$ and latent dimensionality, we find that increasing $β$ monotonically increases sparsity and reduces average spiking activity in the P-VAE. In contrast, Grelu-VAE representations remain unchanged, confirming that the effect is specific to Poisson statistics rather than a byproduct of non-negative representations. These results establish Poisson variational inference as a promising foundation for a resource-constrained theory of computation.

翻译：生物系统中的计算本质上是受能量约束的，然而标准的计算理论却将能量视为可自由获取的资源。本文认为，在泊松假设下的变分自由能最小化，为构建能量感知的计算理论提供了一条原则性路径。我们的关键观察是，泊松自由能目标函数中的Kullback-Leibler (KL)散度项变得与模型神经元的先验发放率成正比，从而产生了一个惩罚高基线活动的涌现代谢成本项。这种结构将一个抽象的信息论量——*编码率*——与一个具体的生物物理变量——*发放率*——耦合起来，使得编码保真度与能量消耗之间可以进行权衡。这种耦合在泊松变分自编码器（P-VAE）中自然产生——这是一种受大脑启发的生成模型，它将输入编码为离散的脉冲计数，并在特殊情况下恢复为一种脉冲形式的*稀疏编码*——但在标准的高斯VAE中却不存在。为了证明这种代谢成本结构是泊松公式所独有的，我们将P-VAE与Grelu-VAE进行了比较，后者是一种对潜在样本应用了ReLU整流的高斯VAE，这控制了非负性约束。通过对KL项加权系数$β$和潜在维度进行系统性扫描，我们发现增加$β$会单调地增加P-VAE的稀疏性并降低平均脉冲活动。相比之下，Grelu-VAE的表征保持不变，这证实了该效应是泊松统计量特有的，而非非负表征的副产品。这些结果表明，泊松变分推断为构建资源受限的计算理论奠定了一个有前景的基础。