The increasing success of deep neural networks (DNNs) in solving complex tasks on par with human-level performance raises the question of whether artificial and biological neural networks perform specific tasks similarly. While current methods of comparison have identified similarities in their respective intermediate representations, a rigorous theory of algorithms and their physical implementations is indispensable for understanding the computational processes biological neural networks implement. This work proposes mathematical definitions of the terms algorithm and implementation and demonstrates on a toy model how to empirically test whether a physical system implements a specific algorithm. Our conceptual framework thus contributes to the efforts in cognitive computational neuroscience to develop rigorous theories that can link the computational, algorithmic, and implementational levels.

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