Darwinian evolution tends to produce energy-efficient outcomes. On the other hand, energy limits computation, be it neural and probabilistic or digital and logical. Taking a particular energy-efficient viewpoint, we define neural computation and make use of an energy-constrained computational function. This function can be optimized over a variable that is proportional to the number of synapses per neuron. This function also implies a specific distinction between adenosine triphosphate (ATP)-consuming processes, especially computation per se vs. the communication processes of action potentials and transmitter release. Thus, to apply this mathematical function requires an energy audit with a particular partitioning of energy consumption that differs from earlier work. The audit points out that, rather than the oft-quoted 20 W of glucose available to the human brain, the fraction partitioned to cortical computation is only 0.1 W of ATP [L. Sokoloff,
Handb. Physiol. Sect. I Neurophysiol. 3, 1843–1864 (1960)] and [J. Sawada, D. S. Modha, “Synapse: Scalable energy-efficient neurosynaptic computing” in
Application of Concurrency to System Design (ACSD) (2013), pp. 14–15]. On the other hand, long-distance communication costs are 35-fold greater, 3.5 W. Other findings include 1) a
-fold discrepancy between biological and lowest possible values of a neuron’s computational efficiency and 2) two predictions of
, the number of synaptic transmissions needed to fire a neuron (2,500 vs. 2,000).The purpose of the brain is to process information, but that leaves us with the problem of finding appropriate definitions of information processing. We assume that given enough time and given a sufficiently stable environment (e.g., the common internals of the mammalian brain), then Nature’s constructions approach an optimum. The problem is to find which function or combined set of functions is optimal when incorporating empirical values into these function(s). The initial example in neuroscience is ref.
1, which shows that information capacity is far from optimized, especially in comparison to the optimal information per joule which is in much closer agreement with empirical values. Whenever we find such an agreement between theory and experiment, we conclude that this optimization, or near optimization, is Nature’s perspective. Using this strategy, we and others seek quantified relationships with particular forms of information processing and require that these relationships are approximately optimal (
1–
7). At the level of a single neuron, a recent theoretical development identifies a potentially optimal computation (
8). To apply this conjecture requires understanding certain neuronal energy expenditures. Here the focus is on the energy budget of the human cerebral cortex and its primary neurons. The energy audit here differs from the premier earlier work (
9) in two ways: The brain considered here is human not rodent, and the audit here uses a partitioning motivated by the information-efficiency calculations rather than the classical partitions of cell biology and neuroscience (
9). Importantly, our audit reveals greater energy use by communication than by computation. This observation in turn generates additional insights into the optimal synapse number. Specifically, the bits per joule optimized computation must provide sufficient bits per second to the axon and presynaptic mechanism to justify the great expense of timely communication. Simply put from the optimization perspective, we assume evolution would not build a costly communication system and then not supply it with appropriate bits per second to justify its costs. The bits per joule are optimized with respect to
, the number of synaptic activations per interpulse interval (IPI) for one neuron, where
happens to equal the number of synapses per neuron times the success rate of synaptic transmission (below).To measure computation, and to partition out its cost, requires a suitable definition at the single-neuron level. Rather than the generic definition “any signal transformation” (
3) or the neural-like “converting a multivariate signal to a scalar signal,” we conjecture a more detailed definition (
8). To move toward this definition, note two important brain functions: estimating what is present in the sensed world and predicting what will be present, including what will occur as the brain commands manipulations. Then, assume that such macroscopic inferences arise by combining single-neuron inferences. That is, conjecture a neuron performing microscopic estimation or prediction. Instead of sensing the world, a neuron’s sensing is merely its capacitive charging due to recently active synapses. Using this sampling of total accumulated charge over a particular elapsed time, a neuron implicitly estimates the value of its local latent variable, a variable defined by evolution and developmental construction (
8). Applying an optimization perspective, which includes implicit Bayesian inference, a sufficient statistic, and maximum-likelihood unbiasedness, as well as energy costs (
8), produces a quantified theory of single-neuron computation. This theory implies the optimal IPI probability distribution. Motivating IPI coding is this fact: The use of constant amplitude signaling, e.g., action potentials, implies that all information can only be in IPIs. Therefore, no code can outperform an IPI code, and it can equal an IPI code in bit rate only if it is one to one with an IPI code. In neuroscience, an equivalent to IPI codes is the instantaneous rate code where each message is
. In communication theory, a discrete form of IPI coding is called differential pulse position modulation (
10); ref.
11 explicitly introduced a continuous form of this coding as a neuron communication hypothesis, and it receives further development in ref.
12.
Results recall and further develop earlier work concerning a certain optimization that defines IPI probabilities (
8). An energy audit is required to use these developments. Combining the theory with the audit leads to two outcomes: 1) The optimizing
serves as a consistency check on the audit and 2) future energy audits for individual cell types will predict
for that cell type, a test of the theory. Specialized approximations here that are not present in earlier work (
9) include the assumptions that 1) all neurons of cortex are pyramidal neurons, 2) pyramidal neurons are the inputs to pyramidal neurons, 3) a neuron is under constant synaptic bombardment, and 4) a neuron’s capacitance must be charged 16 mV from reset potential to threshold to fire.Following the audit, the reader is given a perspective that may be obvious to some, but it is rarely discussed and seemingly contradicts the engineering literature (but see ref.
6). In particular, a neuron is an incredibly inefficient computational device in comparison to an idealized physical analog. It is not just a few bits per joule away from optimal predicted by the Landauer limit, but off by a huge amount, a factor of
. The theory here resolves the efficiency issue using a modified optimization perspective. Activity-dependent communication and synaptic modification costs force upward optimal computational costs. In turn, the bit value of the computational energy expenditure is constrained to a central limit like the result: Every doubling of
can produce no more than 0.5 bits. In addition to 1) explaining the
excessive energy use, other results here include 2) identifying the largest “noise” source limiting computation, which is the signal itself, and 3) partitioning the relevant costs, which may help engineers redirect focus toward computation and communication costs rather than the 20-W total brain consumption as their design goal.
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