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How does a brain wire itself, starting from a single cell, using only the information encoded in a genome? We pose this as an engineering problem: Write a program that a single cell executes to build itself into a brain. The program must be small enough to fit in a genome, and fast enough to finish within developmental time. A computer scientist who knows little about biology quickly realizes why the obvious strategies, which developmental biology has rejected experimentally, fail at scale: The genome is too small to store per-synapse wiring, and axons searching blindly for targets would take too long. These same algorithmic constraints drive the design toward solutions that resemble the developmental strategies organisms actually use. That the same solution structure falls out of scaling limits as out of evolution suggests that features of neural development can be grounded in computational necessity rather than contingency alone. Where earlier thought experiments reached discouraging conclusions about the ability of a biologist to fix a radio1, or to understand a microprocessor2, the news here is better: The design of the brain may be recoverable, not from taking it apart, but from asking how a genome must specify it in the first place.

Suppose we hand a computer scientist the complete wiring diagram of a brain and ask: Write a program that a single cell executes to elaborate itself into this network. The program must fit in a genome-sized instruction set of about 1 gigabyte, and it must halt within a developmental time window of at most about a year. The challenge is that the algorithm must scale: It should work for the 302 neurons of C. elegans, the neurons of a fly, the neurons of a mouse3, and the neurons of a human brain4 (Fig. 1). How would someone with no knowledge of developmental biology approach this problem?

The computer scientist is, in effect, formalizing a task in the spirit of Turing5, von Neumann6, Waddington7, and Brenner8, who all saw the genome as a set of instructions for making an organism rather than a description of one. Formalizing the task helps to identify the algorithmic strategies that are both feasible and tractable. Working through the exercise, we find that strategies that work for a small nervous system hit hard walls as the network grows, and the constraints narrow the field to a small class of solutions. The convergence between theory and experiment informs both directions. For developmental neuroscientists, it formalizes familiar molecular mechanisms as solutions to a well-defined algorithmic problem and picks out which features of those mechanisms are computational necessities and which are free to vary. For theorists, it presents biological development as a system that solves an as-yet-unsolved computational task. Modern machine learning does not yet know how to generate large functional networks from compact descriptions, relying instead on either a large stored weight matrix copied from disk, or a training set that is orders of magnitude larger9. The initial zygote contains exactly such a description, and analyzing the developmental process may help inspire new algorithms for artificial systems.

The problem and its constraints

What does it mean to write a developmental program10–14? The computer scientist does not get to place neurons and wire them up from the outside, like an engineer soldering a circuit board. Instead, she must write a set of instructions that go into a single initial cell, and then she walks away. This initial cell divides, and each daughter inherits the same program and runs it independently. Each instance of the program can read only what it can sense locally: its own internal state along with molecular signals from its neighbor cells that are themselves running the same program. The cell performs actions that will build the brain: It divides, migrates, extends growth cones that navigate by reading molecular cues and form synapses upon arrival. The entire construction process is distributed and recursive, since every signal a cell reads was produced by another cell executing the same code. The programmer's sole lever is the instruction set she loads into the genome at the start, along with the cell's initial state.

The program needs to yield the correct network while respecting practical constraints. We focus here on two aspects: It needs to be short enough to fit in the genome and finish in a reasonable amount of time. A human brain has neurons, each of which forms connections with as many as targets15, for a total of as many as connections.

The program must satisfy two hard constraints.

The information constraint. The instructions for building the brain must ultimately reside in the genome, which contains approximately base pairs21. Each base pair represents 2 bits, because there are four nucleotide bases. The total information content of the genome is therefore roughly bits, or about bits as a generous upper bound. This bound is generous as it assumes the genome has no redundancies, and that every nucleotide is devoted to brain wiring, leaving nothing for the rest of the organism. Any wiring strategy whose instructions exceed this budget cannot be the one biology uses. The budget does not vary much across species: C. elegans and a human differ by less than two orders of magnitude in genome size but by more than eight in neuron count (Fig. 1). Whatever algorithm wires a mammalian brain must scale sub-linearly with the number of neurons.

The time constraint. The wiring process must complete within a developmental time window. With a bounded growth rate25–27, total axon length and developmental time are proportional: A strategy that requires growing too much axon, either transiently or permanently, also takes too long. Any viable strategy must keep both the total axon length and the total developmental time sub-linear in the number of neurons.

Our computer scientist will consider three approaches over the next few sections, illustrated in Figs. 2–5. The computer scientist evaluates each approach against the problem constraints, until she finds a strategy that does not violate any. For now, she restricts herself to deterministic, activity-independent processes: Each cell executes its program without feedback from neural activity, and is not subject to noise. Plasticity, in particular, operates on connections that already exist; the problem of getting axons to the right populations in the first place is the first one she considers. Stochasticity and activity-dependent plasticity are discussed briefly in later sections.

Strategy 1: the identity approach

In his 1974 Turing Award Lecture28, Donald Knuth stated that premature optimization is the root of all evil †. Honoring his famous maxim, our computer scientist first pursues the most naive algorithm (Fig. 2): Assign every neuron a unique identity label, store in the genome a list of target labels for each neuron, and have each growth cone iterate through its list, navigating to each target and forming a synapse. But how does a growth cone find a cell with a given label? Without any map or spatial structure to exploit, the best a growth cone can do is search systematically, visiting cells until it finds a match.

The problem is analogous to finding a book from an unsorted library. On average, a search turns up the right book after going through half the shelves; in the worst case, all of them. The search cost grows in proportion to the size of the collection. With neurons and no navigable structure, a growth cone must extend past up to cells to find one target. The total axon length per target is therefore , growing linearly with brain size. For a human brain with neurons packed into a volume of about 1,300 cm, some 15 cm across4, the mean cell spacing is 50 m, so an search extends roughly 500 km of axon per target, orders of magnitude beyond what a neuron can produce (and prune) in the weeks-to-months of development27,29.

But axon length is not the only problem. The target list must come from the genome, which is the same in every cell. A neuron's genome must therefore contain not only its own targets, but a wiring list for each of the neurons. Fig. 1 showed that an adjacency matrix encoding incurs an information cost on the order of , or bits. However, this does not exploit the fact that the brain's wiring is very sparse: The vast majority of neuron pairs do not connect. Instead of listing all neuron pairs, it would be much more economical to encode, for each neuron, a list of labels for each adjacent neuron. How many bits does it take to label each neuron uniquely? With 1 bit you can distinguish 2 cells; with 2 bits, 4; with 3 bits, 8. Each additional bit doubles the number of distinct labels, so labeling cells requires bits per label, about 33 bits for neurons. Each of the neurons needs a list of target labels, and each label is bits long. The total required storage is , or bits. For a human brain, that comes to roughly bits, still orders of magnitude beyond the genome's -bit capacity (the upper bound, assuming every bit in the genome was devoted to wiring up the brain, which of course it is not).

Strategy 1 fails both constraints: The time required to generate the requisite axon length is too great and the list of instructions is too large.

Developmental neuroscience arrived at the same conclusion. Sperry's chemoaffinity hypothesis proposed that each neuron carries a molecular identity and finds its target by label matching, the central idea of Strategy 130. Reasoning from regeneration experiments, in which severed axons re-found their original targets, Sperry suspected “literally millions, and possibly billions, of chemically differentiated neuron types, each distinguishable from all others”30. The hypothesis was generative and launched a productive search for identity molecules. That search turned up remarkable molecular diversity (Drosophila Dscam1 generates over 10,000 isoforms31, and vertebrate protocadherins create combinatorial identity codes32), but this diversity mediates local self-avoidance rather than global target selection33,34. The field has since moved from precise lock-and-key profiles toward logical recognition rules implemented in molecular machinery34–36, the same shift the scaling argument favors.

Strategy 2: the guidepost approach

The search problem—finding the right partners efficiently—is the more obvious failure, so the computer scientist attacks it first (Fig. 3). Back in the unsorted library: Instead of wandering the shelves, someone provides a sequence of landmarks. To find Moby Dick, first go to the shelf with War and Peace; from there, look for Pride and Prejudice; from there, find Paradise Lost, which is near your target. Each landmark is a marker the growth cone can recognize when it arrives at that location; the growth cone reads them in sequence along its physical path. The growth cone never searches blindly; it always knows the next step.

This solves the navigation problem. In Strategy 1, without directions, the growth cone had to search cell by cell, producing a path length proportional to the number of neurons . Landmarks turn that exhaustive search into a direct walk: The growth cone proceeds straight toward its target, guided at each step. A directed walk through a three-dimensional volume is much shorter than an exhaustive search through it, because neurons packed into three dimensions are only cell-widths apart. The axon length per target drops from to .

But the storage problem gets worse. The directions are specific to your starting point. Someone starting from a different shelf needs a completely different set of landmarks. Each neuron starts from a different location, so each needs its own set of directions for each target. If each path requires waypoints, the genome must store labels, each costing bits as before; the space complexity is , exceeding Strategy 1 by a factor of . The guidepost approach trades one problem for another: It shortens the path but inflates the instructions.

The guidepost model of axon guidance34,37–39, in which long axonal trajectories are broken into short segments bounded by successive molecular choice points, documents the phenomenon Strategy 2 describes. The model has been enormously productive: It accounts for how axons navigate tissue boundaries, cross the midline, and enter target regions. But observing that an axon turns at a series of choice points does not account for how those points are specified. If each cue is independent, posted in the genome one waypoint at a time, the cost grows with every step, and the scaling argument rules this version out. The cheaper alternative is that the cues are not independent but readouts of an underlying order, the move Strategy 3 makes. Just as a reader who does not know the alphabet cannot tell whether a library is sorted or merely signposted shelf by shelf, the trajectory alone cannot tell the two strategies apart.

The computer scientist is on the right track. Navigating by guideposts works; the problem is that the guideposts are arbitrary. Each one is an ad hoc molecular marker that the genome must specify and the growth cone must be taught to recognize. What the computer scientist needs is a systematic set of guideposts, one where every location has a predictable address.

Strategy 3: the coordinate approach

The obvious first attempt to bring structure to the guideposts is a coordinate grid. Back in the library: Moby Dick is on the third floor, southeast corner, twelve feet from the wall, second shelf (Fig. 4). With this kind of address, any reader can find any book without custom directions for each starting point. But an external coordinate system has a serious flaw: If the library remodels—moves into a taller building, rearranges the floors—every address has to be rewritten. The labels describe positions in an external frame, and when the frame changes the labels break. This is a Cartesian grid with , , -coordinates, and it has the same flaw for the brain. The brain is built from a single cell, the tissue changes shape as it grows, and any address tied to a fixed external frame would be rewritten every time the frame moved.

But there is a deeper reason the Cartesian grid is not just fragile but outright unimplementable. The argument above is written from the outside. It assumes someone is stamping coordinates onto cells and worries that the stamps will move. From the cell's point of view—the only one that matters, since the computer scientist has walked away—there is nothing to stamp with in the first place. A Cartesian address like presupposes a ruler and an origin, and both live outside the cell, even outside the organism. There is no privileged reference frame.

Alphabetical order solves this by providing a reference frame in relative coordinates. No one needs to know which floor or shelf a book occupies. The ordering itself indicates where any book sits relative to any other, without the need to memorize coordinates. The same order that assigns each book a position also enables fast lookup: start near the middle of the rack, check whether your title comes before or after, repeat. Each step halves the remaining shelves. A 1,000-book library requires only about 10 halvings (); a million-book one requires only 20. The number of steps grows as : very slowly, even as gets large. The sorting process builds the coordinate system and the navigation structure in one step. The guideposts are still there—they are the alphabetical identities of the books on each shelf—but they arise from the organization of the library rather than from an external reference frame.

Cell division has the potential to provide this kind of ordering for developing tissue (Fig. 5). Every time a progenitor divides, it produces two daughters. If each daughter inherits a label that encodes its position in the division tree—a binary address, extended by one bit at each division—then after rounds, every cell has a unique -bit address that is lexicographically ordered across space40 (Fig. 6). Consider three rounds of division. The first splits the tissue in two: One daughter is labeled 0

, the other 1

. The second division splits each half again, appending a second bit: 00

, 01

, 10

, 11

. After a third round there are eight cells with 3-bit addresses. Two cells sharing the prefix 01

descend from the same second-round daughter and sit near each other in the tissue. Cells that share a recent ancestor have similar addresses, and the tree mirrors the spatial structure of the tissue. Each bit in the address corresponds to a spatial axis: The first division might separate left from right, the second anterior from posterior, the third medial from lateral. In biological terms, each bit is a molecular distinction between daughter populations—a transcription factor expressed in one lineage but not the other, or a gradient that partitions a domain into two subdomains with different identities41,42. A cell's full address specifies where it sits in the tissue. This coordinate system does not require an external reference; it is inherited at each division, generated for free by the process that creates the cells. The map also continues to grow: Earlier-wired axons serve as scaffolds that later axons follow, extending the navigable structure over time38,43. Wiring rules do not have to be rewritten when the brain changes shape.

The coordinate system can solve navigation: At each branch point the growth cone checks one bit of its target address and picks the corresponding half of the tissue, reading the full address in bit-comparisons. The physical path length is still , the same as Strategy 2; the win over Strategy 2 is in information cost, not time. It also provides a natural language for writing wiring rules.

The simplest rule in this language is “connect to your four nearest neighbors (north, south, east, west),” but that generates only a grid. Structured addresses allow richer rules. A single bit flip in the address can connect a neuron to its contralateral partner, on the opposite side of the brain. A different bit flip can connect it to a cell in a different region or a different layer. Different cell types can carry different rules, enabling the formation of complex circuits44.

The mechanics are straightforward. Because each bit in a cell's address encodes a position in the developmental hierarchy, two cells whose addresses differ in a specific bit position sit on opposite sides of that division. Cells differing in bit 1 are contralateral partners. Cells differing in bit 2 are anterior-posterior partners. So “connect to your contralateral partner” translates to “flip bit 1.” The rule can be written as a bit mask, a binary pattern specifying which bits to flip (Fig. 6). The mask 101

applied to any source address flips the first and third bits, producing the target address. The genome stores this mask once, and every neuron of that type applies it to its own address. The specific transformation is schematic, and real mappings involve multiple molecular signals. The key property still holds: The rule is shared across all neurons of a type, and its length scales as , not .

Like any language, this one makes some things easy to say and others hard. Contralateral projections, topographic maps, laminar targeting, and region-to-region connections can all be cheaply expressed as transformations on developmental addresses42,45. But a rule like “connect to these 47 arbitrary neurons scattered across the brain” would be just as expensive as in Strategy 1 (but still cheaper than Strategy 2). The coordinate system is a bet that most of the brain's wiring can be described as compact transformations on addresses. If so, the genome's budget is sufficient. If not, no coordinate system can help.

The numbers bear this out. Each rule is bits long, about 33 bits for neurons. With cell types46 and targets per type, the total storage is bits, comfortably within the genome. By contrast, Strategy 1's per-neuron target lists would require bits, about 300,000 times the genome's capacity, and Strategy 2 is worse by a factor of waypoints per path. The path length per target is for both Strategies 2 and 3; what separates them is information.

The binary addresses correspond to combinatorial patterns of transcription factors and signaling molecules that distinguish one brain region from another. The first division of the neural tube into left and right halves is one “bit”, maintained by differential gene expression across the midline47. Subsequent subdivisions along the anterior-posterior and dorsal-ventral axes add further bits, partitioning the tissue into discrete domains that differ in the transcription factors they express41,48,49. A growth cone navigating this hierarchy reads molecular markers at each boundary, the biological equivalent of checking a bit at each branch point. The resulting coordinate system is not transient: Cells retain a heritable, cell-autonomous memory of their position long after the original patterning signals are gone50, maintained by epigenetic mechanisms such as differential methylation51. This positional memory has functional consequences for wiring: Denervated muscles are preferentially reinnervated by motor axons from positionally matched spinal segments52,53. The coordinate system installed by development becomes a permanent molecular property that axons can read.

The hierarchy likely arises from multiple overlapping mechanisms48,54: morphogen gradients acting upstream of gene regulatory networks49,55–58, and potentially cell lineage itself42,59–62. Recent work suggests lineage may play a direct role: A cell lineage tree can install a global space of molecular addresses that axons read to navigate40, and the predicted hierarchical structure appears in brain-wide gene expression patterns in both mouse and zebrafish63.

Strategy 3 makes a strong prediction. If the brain wires itself using an address system, then that address system is a real map, written in gene expression and spread across the whole brain. It is laid down early and stays in place into adulthood. Two things follow. First, the map should be visible: You should be able to read it off gene expression in an adult brain63. Second, a small set of rules applied to that map should predict which neurons connect to which40.

The role of noise

The problem as formulated so far has been to write a program that generates a specific wiring diagram. That is a fair approximation of the problem statement for C. elegans, whose 302 neurons and roughly 7000 synapses are wired almost identically from animal to animal22,64. But as brains get larger, development allows stochastic outcomes and no longer specifies a single diagram, but rather a distribution65,66. Even the brains of identical twins are not wired alike67,68. Evolution selects for rules whose distribution is mostly good: Most wiring diagrams sampled from it should represent working brains.

Noise can make algorithms faster and simpler69. Suppose every neuron in area A needs to sample about 10 inputs from area B, and each cell in A should get an uncorrelated subset. This goal is achieved easily with the rule “connect to each candidate target with probability ,” for candidate targets in B. Each neuron ends up with a different random subset, which is what the circuit requires. The rule is cheap because the thermal and molecular fluctuations it draws on are always present. A deterministic rule like “connect to the closest 10 neurons in B that make an uncorrelated set of inputs” also fulfills the brief. However, without stochastic choices, the neurons in A would have to explicitly negotiate with their neighbors to ensure all sets are uncorrelated. Deterministically removing correlations incurs an overhead that is avoided by the stochastic algorithm, which uses noise as a free resource. In general, noise facilitates algorithms that require independent variables.

But noise is no silver bullet. The same noise that makes variability cheap can also make consistency expensive. Area A can randomly sample 10 inputs from area B, but left unchecked, noise could accidentally sample from area C, or sample too few or too many inputs. A small error left uncorrected in early development often proves fatal at later stages. Von Neumann70 and Waddington7 both considered how noisy processes could yield reliable results. Discretization emerged as a solution for both computers and organisms. A computer's memory has two discrete bins: 0 and 1. The voltage can fluctuate noisily within a bin, but as long as it does not cross the threshold that separates them the calculation is unaffected. Waddington's canalization rests on the same principle. Developmental trajectories are sorted into a discrete number of canals. Noisy variation within a canal yields the same developmental output, as long as the trajectory does not go beyond the canal's banks. The cost of discretization lies in the thresholds and banks. The banks need to be specified in the genome, and higher banks that are robust to heavier fluctuations incur higher costs71,72.

Specifying distributions of wiring diagrams rather than a specific instance has a potential advantage. It can lead to more compact rules. Rules that leave open more possibilities often need less information than those that pin down specifics73,74. A rule that prescribes a specific neuronal target—connect neuron to target neuron located in area X—requires more information in bits than one that merely says “navigate to area X and connect to any partner of type Y.” Whether distributions and stochasticity are advantageous or detrimental depends on the particular algorithm and use-case. Evolution may have found the special cases and algorithms that exploit the advantages, and minimize the disadvantages.

Plasticity

Plasticity refines the wiring after the initial rules have run. There are two forms of plasticity. One form, based on spontaneous activity, only uses information that is ultimately derived from the genome. It thus fits neatly into the problem defined in the introduction, namely to find a program that builds a wiring diagram from instructions that fit in a genome. The other—experience-dependent plasticity—goes beyond that problem, as it imports information from the outside world, beyond that found in the genome.

The internal form narrows a distribution that the initial wiring left too broad. An architect draws plans precise enough to place the walls and plumbing, but the contractor adjusts doors and trim on site, because no blueprint anticipates exactly how the lumber warps or the foundation settles. Retinal waves, synaptic competition, and pruning play the contractor's role75–77. The genome specifies a rule such as “strengthen co-active synapses,” or “prune the weakest connections.” Activity determines which specific pattern satisfies it. During development this activity can be generated internally. Retinal waves sweep across the retina before the eyes open75. The pattern the circuit settles into is still one the genome specified, a competition-based rule of the kind discussed above. Feedback of this kind can change how long development takes, or how reliably it converges, but it adds no information beyond what the genome already holds. The circuit refines toward a genome-specified answer using a signal it generates for itself.

Experience-dependent plasticity draws on information from outside the organism. The genome does not contain the vocabulary of one's native language, the faces of one's family, or the locations of fruit trees. Natural selection would not store this information even if it could, because it varies faster than evolution can track. The genome can store priors that make the information easier to absorb, such as the general structure of human language or of natural images78. The genome supplies experience-dependent rules that run on data supplied by the environment, similar to how the stochastic rules from the previous section run on noise supplied by the cell. The connectome results from a genomic program that runs against both the noise it encounters and the environmental input it is exposed to.

The two forms look alike because the machinery is often shared. The plasticity that provides corrective feedback during development is frequently the same apparatus that later supports learning79,80. Their informational roles differ: One converges on an answer the genome already implies, the other fills actual blanks the genome left open. The shared mechanism reflects a shared evolutionary origin. The genome did not invent learning from scratch. It already had a system that adjusted connectivity in response to activity, used to refine circuits against internally generated waves. Repurposing that system for experience-dependent learning may have taken little new molecular apparatus, mainly a change in what supplies the reference point, from spontaneous activity to the sensory world.

Discussion

Two influential thought experiments have asked whether the methods of biology could recover the design of a system whose ground truth is known. Lazebnik asked whether a biologist could fix a radio1; Jonas and Kording asked whether a neuroscientist could understand a microprocessor2. Both came back with the same discouraging answer. Lesion studies, activity correlations, and dimensionality reduction failed to recover the logic of even these simple engineered systems. These thought experiments argue that since these methods cannot recover the logic of a radio or a microprocessor, there is little reason to expect them to explain the brain from its wiring and its activity.

The problem is not confined to analogies with radios and microprocessors. Understanding a trained artificial neural network is an important instance of the same hard problem, addressed by a subfield of artificial intelligence, called “mechanistic interpretability.”81 In artificial networks synaptic weights and activations are known. Yet progress in explaining how the neurons and circuits inside trained artificial networks compute has been slow. Using neuroscience-inspired approaches, researchers have, for example, identified populations of units in a large language model (LLM) that represent the Golden Gate Bridge; artificially activating those units, an intervention reminiscent of optogenetics, pushed the model to steer almost any conversation toward the bridge, yielding many amusing conversations82. But even with complete access to essentially unlimited amounts of data about weights and activations, these methods have yielded only modest progress in understanding how a language model comes to behave as such an impressive simulacrum of human thought, or how to reshape one to be more trustworthy and reliable.

Artificial and biological nets have different description lengths. The recipe for an artificial neural network seems simple: an objective, a learning rule, an architecture, and a data set. But the data set is the training data, which for modern LLMs contains most of the internet9. The act of training compresses these data into a list of weights. The biological analogy to this process is not experience-dependent learning, but evolution, which captured and compressed millions of years of data78. But rather than a long Strategy 1-type list, evolution stored a much shorter developmental program into a genome, which produces those weights when run83. We have had limited success in reading the behavioral and computational strategies that gradient descent and evolution gleaned from their data by inspecting weights and activations. Reading those strategies from the much shorter developmental program may be an easier problem.

Decades of molecular genetics have dissected the developmental program—the morphogens, the transcription factors, the guidance cues and their receptors84,85—in remarkable detail. Here we take a step toward formalizing an algorithmic theory: an account of how a genome specifies brain-wide connectivity within explicit information and time budgets, and of which mechanisms those budgets force. We started with two fundamental constraints: the information a genome can store, and the time development is allowed to take. Working through that exercise, we found the two constraints demanding. Most ways of specifying a brain fail one or the other. The strategies that survive fall into a narrow class: hierarchical coordinates laid down by cell division, wiring rules written as compact transformations on those coordinates, and each rule stored once per cell type rather than once per neuron86. This is close to what development appears to do. Viewed through these constraints, the molecular mechanisms worked out over decades can be cast as steps in an algorithm. For any given mechanism, we can then ask whether the constraints force it, or leave it free to vary.

Evolution, development, and learning are one process operating across multiple time scales. Evolution compresses the experience of a lineage into the genome over hundreds of millions of years. Development unfolds that compressed record into a network over the course of hours to years. Learning fine-tunes the result within a single life. The genome is where the long experience is banked; development is how it is unfolded; plasticity is the fast correction when the world moves faster than evolution can track. To read the developmental program as an algorithm is to see how experience, at every scale, becomes structure.

Notes

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