Title: Bounds on the computational complexity of neurons due to dendritic morphology
Venue: NeurIPS 2025
Accepted: yes
Keywords: computational neuroscience

dendritic integration

neuroAI

neuromorphic computing

systems neuroscience
Abstract: The simple linear threshold units used in many artificial neural networks have a limited computational capacity.  Famously, a single unit cannot handle non-linearly separable problems like XOR. In contrast, real neurons exhibit complex morphologies as well as active dendritic integration, suggesting that their computational capacities outperform those of simple linear units.  Considering specific families of Boolean functions, we empirically examine the computational limits of single units that incorporate more complex dendritic structures.  For random Boolean functions, we show that there is a phase transition in learnability as a function of the input dimension, with most random functions below a certain critical dimension being learnable and those above not.  This critical dimension is best predicted by the overall size of the dendritic arbor. This demonstrates that real neurons have a far higher computational complexity than is usually considered in neural models, whether in machine learning or computational neuroscience.  Furthermore, using architectures that are, respectively, more "apical" or "basal" we show that there are non-trivially disjoint sets of learnable functions by each type of neuron.  Importantly, these two types of architectures differ in the robustness and generality of the computations they can perform.  The basal-like architecture shows a higher probability of function realization, while the apical-like architecture shows an advantage with fast retraining for different functions.  Given the cell-type specificity of morphological characteristics, these results suggest both that different components of the dendritic arbor as well as distinct cell types may have distinct computational roles. In single neurons, morphology sculpts computation, shaping not only what neurons do, but how they learn and adapt. Our analysis offers new directions for neuron-level inductive biases in NeuroAI models using scalable models for neuronal cell-type specific computation.

Bounds on the computational complexity of neurons due to dendritic morphology

Abstract

The simple linear threshold units used in many artificial neural networks have a limited computational capacity. Famously, a single unit cannot handle nonlinearly separable problems like XOR. In contrast, real neurons exhibit complex morphologies as well as active dendritic integration, suggesting that their computational capacities outperform those of simple linear units. Considering specific families of Boolean functions, we empirically examine the computational limits of single units that incorporate more complex dendritic structures. For random Boolean functions, we show that there is a phase transition in learnability as a function of the input dimension, with most random functions below a certain critical dimension being learnable and those above not. This critical dimension is best predicted by the overall size of the dendritic arbor. This demonstrates that real neurons have a far higher computational complexity than is usually considered in neural models, whether in machine learning or computational neuroscience. Furthermore, using architectures that are, respectively, more 'apical" or 'basal" we show that there are non-trivially disjoint sets of learnable functions by each type of neuron. Importantly, these two types of architectures differ in the robustness and generality of the computations they can perform. The basal-like architecture shows a higher probability of function realization, while the apical-like architecture shows an advantage with fast retraining for different functions. Given the cell-type specificity of morphological characteristics, these results suggest both that different components of the dendritic arbor as well as distinct cell types may have distinct computational roles. In single neurons, morphology sculpts computation, shaping not only what neurons do, but how they learn and adapt. Our analysis offers new directions for neuron-level inductive biases in NeuroAI models using scalable models for neuronal cell-type specific computation.

1 Introduction

The relationship between neuronal structure and function has received renewed attention due to the growing appreciation of morphoelectric diversity across cell types [Gouwens et al., 2019, 2020, Dembrow et al., 2024, Kalmbach et al., 2021]. Yet, other than classical studies of synaptic integration and circuit motifs, the computational roles of single neurons, shaped by their dendritic morphology, remain underexplored. While traditional models often simplify neurons as linear threshold units, such approximations neglect the rich, nonlinear processing capabilities arising from the spatial and electrical organization of dendritic arbors. Recent experimental evidence demonstrates that

dendrites actively integrate inputs in a manner that is both cell-type specific and nonlinear [London and Häusser, 2005]. These findings suggest that dendritic and morphological properties endow neurons with complex computational capabilities, positioning them between simple linear classifiers and deep artificial networks [Cazé et al., 2013, Bicknell and Häusser, 2021].

It has been shown that single neurons can transcend passive linear integration and compute nonlinearly separable functions, such as the XOR function [Gidon et al., 2020] and feature-binding tasks [Bicknell and Häusser, 2021]. However, these studies typically focus on specific nonlinearly separable problems and therefore do not quantify the broader computational limits of single neurons across a range of function complexities. Prior work [Cazé et al., 2013] demonstrated that neurons with local nonlinearities can leverage different integration strategies to solve certain nonlinearly separable tasks. Building on this foundation, we employ Boolean functions to systematically scale the complexity of computational problems with the dimension of input. In addition, we explore specific families of Boolean functions to characterize the capabilities of the dendritic arbor.

In this work, we examine how dendritic morphology constrains the upper bounds of computational complexity achievable by a single neuron. We model neurons as abstract networks of non-linearly integrating dendritic compartments, organized according to archetypal branching architectures. Within this framework, dendritic networks are tasked with learning Boolean functions, allowing precise control over input complexity and objective difficulty. By assessing function realizability across different input dimensions and measures of Boolean function complexity (here, entropy [Mingard et al., 2019] and sensitivity [Kenyon and Kutin, 2004]), we systematically relate morphological features, such as breadth and depth of dendritic trees, to the space of computations that single neurons can perform.

To ground this analysis, we compare two canonical morphologies: (1) a shallow, broad topology reminiscent of basket cells and basal dendritic tufts [Häusser and Mel, 2003, Tzilivaki et al., 2019], and (2) a deep, narrow topology modeled after the apical tufts of excitatory neurons [Häusser and Mel, 2003, Spruston, 2008]. Despite containing the same number of nonlinearly integrating compartments, these architectures exhibit distinct computational strengths: shallow networks excel at learning low-dimensional, low-sensitivity functions, while deep networks generalize more robustly and retrain more efficiently across complex tasks. Importantly, we show that dendritic architectures impose a critical input dimensionality threshold, beyond which the probability of learning a randomly sampled Boolean function collapses sharply. This behavior is reminiscent of phase transitions in circuit complexity (e.g., phase transition in 3-SAT, Karp [2009]) and highlights the intrinsic computational limits imposed by morphology. Furthermore, different dendritic architectures preferentially realize distinct subsets of Boolean functions, suggesting a possible division of computational labor across cell types within neuronal ensembles. Beyond static realizability, we examined how morphology shapes the dynamics of learning itself. We show that apical-like architectures, though less robust initially, retrain and generalize more efficiently once pre-trained, reflecting a flexibility-robustness trade-off that may underlie how different cell types contribute to adaptive computation. Finally, extending this framework, we adapt these morphological motifs to transformer architectures, revealing analogous trade-offs between broad (basal-like) and hierarchical (apical-like) attention patterns, thus bridging biological and artificial systems. By abstracting dendritic trees into structured, deep network-like architectures, our work offers a biologically grounded and tractable framework for understanding how single-neuron computations are shaped, and limited, by morphology. Incorporating such cell-type-specific computations into models of neural circuits opens new avenues for rethinking both theoretical and systems neuroscience paradigms.

2 Testing the limits of single neuron computation with abstract models of dendritic morphology

Abstract Model of a Nonlinear Dendritic Network

Wemodel a single neuron as a feedforward network comprising dendritic rectification units arranged according to the neuron's dendritic arbor (Fig. 1A). Each compartment b acts as a nonlinear unit in an equivalent multilayer representation:

Soma

inputs to dendrite

Soma

(A)

(B)

Shallow and broad

dendritic circuit

Deep and narrow

dendritic circuit

Soma

Dendritic

branches

x

1

x

2

x

3

x

n

n-dimensional

input layer

Every unit sums inputs from input layer

and the summed input undergoes

a non-linear transformation:

(a) 'Basal' like

(b) 'Apical' like

where y b is the output of branch b , h b and θ b are branch-specific gain and threshold parameters, W b i are synaptic weights, and σ is the logistic sigmoid. If a branch b ′ also receives inputs from other branches b ,

The soma rectifies its dendritic and direct inputs:

where ∆ is the somatic threshold. Fig. 1A shows the corresponding network schematic, with inputlayer skip connections indicating that all compartments receive direct input.

To test how dendritic arrangement affects computation, we considered two morphological extremes representing trade-offs in circuit breadth and depth under limited total compartments. The basal/parallel model has a single hidden layer where all nonlinear compartments connect directly to the soma, resembling basal tufts or basket cells [Tzilivaki et al., 2019, Spruston, 2008]. The apical model organizes the same number of compartments into a hierarchical tree, resembling apical tufts of excitatory neurons [Spruston, 2008]. Such an equivalent architecture, if constrained by the number of available dendritic compartments, will be deeper but narrower than the basal model. We typically compared both models with a small dendritic circuit size, N branch = 7 compartments (Fig. 1B).

For comparison, we implemented a linear integration model in which all branches sum inputs linearly before a final somatic nonlinearity:

All models, including the linear baseline, used identical input dimensionality and total trainable parameters, differing only in structural organization and presence of intermediate nonlinearities.

Characterizing the Computational Complexity of a Dendritic Network The computational complexity of a Boolean function, given a fixed gate set such as AND and OR, is characterized by the number of gates and the minimum circuit depth required to implement it [Wegener, 1987]. Here, we ask: given a specific multilayer network, what limits its ability to realize a family of Boolean functions?

A single neuron receives N dim input streams that may represent distinct information sources. An input X i = 0 denotes an inactive, noisy source, and X i = 1 an active one; equivalently, 0/1 can be viewed as small versus large EPSPs from distinct synaptic inputs.

Previous work on dendritic learning with increasing input size has focused on memory capacity [Poirazi and Mel, 2001]: the number of patterns learnable within a given N dim . Here, we test instead the ability of a dendritic circuit to implement arbitrary input-output mappings defined by Boolean truth tables, providing a direct measure of computational rather than storage capacity. As shown by Gardner and Derrida [1988], the number of patterns learnable by a linear perceptron scales as p = aN dim with a < 1 . Hence, as N dim increases, the fraction of linearly separable functions f = p/ 2 2 N dim rapidly approaches zero, reflecting the rise in functional complexity with dimensionality. Thus, N dim serves as an order parameter for scaling task difficulty.

For dimension N dim , there are x N dim = 2 N dim possible inputs and n func = 2 2 N dim total Boolean functions. Because testing all Boolean functions in high dimensions ( N dim > 4 is infeasible, we define three representative families that span the space of task complexity. First, the class of typical functions: randomly sampled outputs where each row of the truth table returns 1 or 0 with equal probability ( p = 0 . 5 ). These typical Boolean functions tend to have high sensitivity (defined in the subsequent paragraphs). In the face of extremely large number of possible functions in input dimensions higher than 4, the class of typical functions is to sample the space of 'common" functions. We evaluate basal-like, apical-like, and linear-like architectures on sampled typical functions across N dim .

Second, we consider the class of Boolean functions of fixed entropy, defined simply as the fraction min(number of input rows that map to 1, number of input rows that map to 0)/total number of input rows in the truth table [Mingard et al., 2019]. For each input dimensionality, we randomly sampled functions with varying probabilities of outputting 1, thereby covering a range from low-entropy (biased) to high-entropy (balanced) functions.

Finally, we define functions of fixed sensitivity [Rubinstein, 1995, Mossel et al., 2005], where sensitivity quantifies output instability to single-bit flips: The sensitivity of a Boolean function in dimension n is defined as follows:

Definition 2.1 (Def 1) . Let f : 0 , 1 n → 0 , 1 be a Boolean function and let x ∈ 0 , 1 n . The sensitivity of f at x is the number of positions i ∈ [ n ] such that flipping the i th bit of x changes the output f ( x ) . The sensitivity s ( f ) of f is the maximum sensitivity s ( f, x ) over all points x .

Sensitivity increases monotonically with entropy (Supplementary Fig. 5) and scales the difficulty of learning for linear architectures (Fig. 3A).

A low-entropy or low-sensitivity Boolean function can be viewed as one containing non-influential input bits that do not alter the output [Kalai, 2016]. Such functions resemble neural computations in which many inputs are high-dimensional but only a subset carries relevant information, making them effectively lower-dimensional. Because sensitivity increases monotonically with entropy, we used the same set of functions with varying entropy to compute sensitivity, and verified through rejection sampling that both approaches produce comparable distributions.

Together, the random, fixed-entropy, and fixed-sensitivity families span tasks from highly structured to maximally random. For each N dim = 3 -8 , we trained all architectures on identical sampled sets (100-500 functions per class) to ensure a fair comparison. Distributions of entropy and sensitivity across function classes are shown in Supplementary Fig. 6.

ABoolean function is considered learned by an architecture if appropriate weights W b i and nonlinear parameters θ b , h b can be optimized to map the input ¯ X = X 1 , X 2 , ...X N dim to its truth table output Y i ∀ i ⊆ 1 , 2 , .... 2 N dim , where i denotes the index of the row of the truth table representing the Boolean function. We quantify learning using the Hamming distance between the (thresholded and binarized) dendritic network output and target (see Supplementary Fig. 1). For each function class with N func functions trained over n trials random initializations, we compute three learning metrics to test the learnability of the class:

The mean Hamming distance between the network output and training target over trials,

The mean and variance of this quantity over the sampled functions N func is then visualized.

Function learning probability,

is the probability of learning a particular random function and n HD =0 is the number of trials where the given function was learned with a Hamming distance ∼ 0 in n HD 0 . This is the fraction of times the network learned the function perfectly.

The 'best-case' learning outcome, i.e. the probability of typical function learning is instead calculated as

where n best HD =0 is the number of functions with a Hamming distance of 0 in at least one of the trials tested. This is the fraction of functions for which there is at least one successful trial.

Training and Optimization Details We used two complementary optimization schemes: gradient descent and non-gradient based method (DIRECT), to ensure that results were not dependent on the learning algorithm. For all training instances, which entailed learning the function mapping f : { 0 , 1 } n →{ 0 , 1 } in input dimension n , the input consisted of the full enumeration of 2 n input rows, with the target being the Y-column of the Boolean function mapping.

While training using the gradient-descent based approach, we used Binary Cross-Entropy as the loss function between the predicted and the target outputs during the training. We used an Adam optimizer with a fixed learning rate, with learning rates and epochs tuned according to the input dimension to ensure convergence of loss (ranging from 6e4 epochs with lr = 5e-3 for N dim = 3 to 6e6 epochs, lr = 1e-4 for N dim = 8 . We also ran instances of training with an Adam optimizer with a learning rate scheduler, but that did not impact realizability trends shown in the paper. Parameters ( W b i , h b , θ b , ∆ ) were initialized from uniform distributions within bounds chosen to match comparable dynamic ranges across architectures.

To verify that the observed realizability trends were not artifacts of backpropagation, we repeated the analyses using the DIRECT global optimization algorithm [Jones et al., 1993, Gablonsky and Kelley, 2001], which does not rely on gradient information. DIRECT performs a deterministic partition-based search over bounded parameter domains until convergence or a maximum of 5 × 10 4 evaluations. Reasonable bounds on the learnable parameters were first set as input parameters for the optimization algorithm. The parameter space was constrained with weights bounded between -20 and 20, branch gains ( h ) between 0 and 10, thresholds ( θ ) between -1 and 10, and output bias ( ∆ ) between -1 and 10. Both basal and apical cell architectures were trained using mean squared error loss between continuous sigmoid outputs and binary target labels. The DIRECT algorithm performed global optimization without gradients, exploring the 7*d +7 + 7 + 1 - dimensional parameter

space (d input dimensions × 7 branches + 7 gains + 7 thresholds + 1 bias) to minimize classification error with a maximum of 2,000,000 function evaluations and zero volume tolerance for convergence.

In both cases, training performance was evaluated using Hamming distance between binarized predictions (threshold 0.5) and ground truth binary labels. We trained the 3 dendritic networks (basal, apical, linear) on Boolean functions with input dimensionality varying from N dim = 3 to N dim = 8 . The training time for a given architecture and function over one set of initial conditions (trial) took a maximum of ∼ 4 hrs ( N dim = 8 ). For multiple trials and functions, we use parallel computing over a cluster to run training on multiple functions and trials simultaneously.

3 A phase transition in functional realizability

(A)

(B)

(C)

We first examined how realizability scales with input dimensionality, by sampling n func = 1000 per input dimension. As expected, a purely linear circuit with only a somatic nonlinearity fails to solve most functions even at N dim = 3 (Supplementary Fig. 2), whereas both basal and apical types perform well in this regime. To mitigate stochastic effects, we trained n func = 100 typical functions across n trials = 10 random initializations. The mean Hamming distance across trials (Fig. 2A) increases sharply with N dim , with the linear model failing earliest and both nonlinear architectures maintaining lower error until higher dimensions.

Fig. 2B-C show the corresponding probability of realizability and best-case realizability for typical functions. All architectures exhibit a sharp transition from high to near-zero realizability, analogous to phase transitions in constraint-satisfaction problems. The linear model transitions around N dim = 4 , while basal and apical architectures remain functional until N dim ≈ 7 . This difference highlights how even limited dendritic nonlinearities delay the onset of intractability.

Although a sufficient number of nonlinear compartments ( N branch ) in dendritic architecture 1(a) can, in theory, approximate any function [Cybenko, 1989], real neurons are constrained by finite compartment number. We find that beyond a critical dimension, the probability of realizing random functions collapses steeply for the basal architecture. This transition was robust to the optimization method: substituting backpropagation with the global DIRECT optimizer [Jones et al., 1993, Gablonsky and Kelley, 2001] produced the same pattern (Supplementary Fig. 4). Beyond this point, functions become effectively unsolvable within a fixed-size nonlinear circuit.

In Supplementary Fig. 3, we analyze how increasing dendritic compartments affects computational capacity. For the basal architecture, size increases by adding branches; for apical, we sample 2-layer tree-like circuits with increasing nodes. In both cases, capacity scales exponentially with compartment number. Because connections are feedforward and sparse, deeper networks trade off breadth, explaining why apical and basal architectures show similar size-cost trade-offs for increasing input dimensionality.

The apical architecture shows lower typical function realizability in low dimensions and greater variability across functions (Fig. 2B), though it matches basal performance in higher dimensions. This

may stem from sensitivity to initial conditions, as uniform weight initialization may disadvantage hierarchical structures. The best-case analysis (Fig. 2C) compensates for this, revealing similar overall trends. We examine this variance further in the next sections.

For some Boolean functions, mismatches in output may be tolerable, e.g., if certain inputs are irrelevant or later corrected by additional signals. Given the high dimensionality of real dendritic inputs, partial learning may suffice. To test this, we relaxed the learning criterion to allow limited output mismatches (Supplementary Fig. 4) and found that in higher dimensions, the apical architecture shows slightly better approximate learning than the basal type.

It is important to distinguish between trainable realizability and best-case realizability . The former reflects functions successfully learned through gradient-based or DIRECT optimization, whereas the latter quantifies the theoretical capacity of the architecture regardless of optimization path. Across both metrics, the phase transition in realizability remained consistent, confirming that the phenomenon is independent of the training algorithm. Some realizable functions remain difficult to learn in practice due to non-convex loss landscapes or limited parameter resolution; constraints that may be biologically meaningful, as real neurons also compute within restricted parameter regimes. Thus, our empirical results capture both the intrinsic computational capacity of a morphology and the accessibility of that capacity under plausible learning dynamics.

4 Morphology-based limits on single-neuron computation

As observed in the error bars in Fig. 2B, sampled functions within a given input dimension N dim can differ substantially in realizability. To capture this variability and differentiate between functions in the same N dim , we require an additional order parameter sensitivity - which quantifies function difficulty beyond dimensionality alone. To broaden the tested function space, we varied entropy, thereby sampling Boolean functions with diverse sensitivities (definitions in Methods). Supplementary Fig. 6 confirms that entropy-based sampling yields a wide distribution of sensitivities, extending beyond the 100 typical functions used earlier.

Fig. 3B compares learning probabilities for 1000 sampled functions by sensitivity across architectures. Apical and basal models specialize in distinct function classes: functions above the diagonal are better learned by apical cells, while those below favor basal cells. Binning by sensitivity (Fig. 3C) shows that basal cells excel at low-sensitivity functions but degrade as complexity increases, whereas apical cells maintain more uniform, though generally lower, performance across sensitivities. This pattern suggests that apical architectures generalize broadly but imprecisely, while basal architectures learn robustly but with limited transfer. We hypothesize that the apical architecture prioritizes generalization before fine-tuning, whereas the basal architecture learns functions independently and more deterministically. We test this hypothesis in the next section.

5 Morphology-based trade-off between computational robustness and flexibility

The apical architecture exhibits lower realizability under random initialization, particularly for lowsensitivity functions. To test whether this reflects distinct learning strategies, we compared retraining flexibility (Fig. 4). Both architectures were first trained on n func = 50 Boolean functions at N dim = 5 , with entropy increasing by 0.1 across batches of 10. N dim = 5 was chosen to be close to the realizability threshold to ensure learnability. For each function, weights from a successful trial ( HD = 0 ) were stored and used for retraining. Each model was then retrained on function i using weights from function j , tracking HD j → i ( t ) over epochs t . The diagonal elements ( j = i ) are zero by design, but pretraining on low-entropy functions ( j = 1 -10 ) accelerates learning for other functions, more strongly in apical than basal architectures, which converges faster than basal across all j (Fig. 4A,B).

We next retrained while fixing branch weights. Branches were indexed by distance from the soma: trunk ( b = 0 ), proximal ( b = 1 , 2 ), and distal ( b = 3 -6 ). Holding the 4 distal branches fixed minimally affected apical retraining but markedly slowed basal retraining, where all branches are equivalent. Additional tests fixing both weights and nonlinearities ( h b , θ b ) confirmed that only proximal branches require tuning for apical re-learning (Fig. 7). These findings suggest that apical cells generalize using distal branches trained on simple functions (e.g., AND/OR), which serve as

(A)

(B)

(C)

(D)

(E)

Sensitivity

Entropy

Hamming Distance

reusable building blocks for more complex tasks. In contrast, basal cells lack this generalization and must relearn each function independently (Fig. 4C). Thus, apical architectures are more flexible but less robust, while basal architectures are robust but less adaptable.

6 Extending morphological motifs to transformer architectures

To test whether the trade-offs observed in dendritic architectures extend to artificial systems, we adapted the 'basal" and 'apical" motifs to the layout of attention heads within a transformer model (Supplementary material 1). Both variants contained comparable parameter counts but differed in how attention heads were arranged; broad and shallow versus hierarchical and deep. Across four sequence-learning tasks (copying, reversal, sorting, and pattern completion), broad configurations achieved higher final accuracy, whereas hierarchical configurations converged faster and showed greater improvement during transfer learning. These results suggest that structural organization alone can impose complementary regimes of robustness and flexibility, paralleling the retraining behavior observed in biological dendritic networks.

7 Discussion

Our results demonstrate that dendritic morphology is not merely a structural feature but a fundamental determinant of a neuron's computational capacity. By abstracting dendritic trees into simple

Increasing number of epochs

Overall space of weights

Space of weights spanned by solutions

for basal cell type

space of weights spanned by solutions

(of low-dimensional/easy functions)

for apical type

initial conditions during trials

Solution weights for example functions

for basal

Solution space for example functions

for apical

(A)

(B)

(C)

models, we show that neural geometry imposes strict bounds on the input-output functions a single neuron can realize. While nonlinear dendritic integration greatly expands computational capabilities beyond linear models, this expansion is not unbounded. Beyond a critical dimension, the probability of learning random Boolean functions collapses. This phase transition mirrors phenomena in circuit complexity theory, such as the 3-SAT satisfiability transition.

Importantly, dendritic architectures have computational trade-offs. 'Basal-like' shallow, broad architectures favor robust, condition-insensitive computation, reliably learning simple functions across diverse initializations. In contrast, 'apical-like' deep, narrow architectures favor flexibility: though initially less robust, once properly initialized, they retrain and generalize across a broader range of functions with fewer adaptations. These complementary properties suggest a natural division of computational labor across cell types: robust processors for stable operations and flexible processors for dynamic reconfiguration.

Neuroscience implications Our results complement and extend a growing body of experimental and theoretical work showing that dendritic morphology shapes how neurons compute. Human layer 2/3 pyramidal neurons exhibit graded calcium-mediated dendritic spikes that enable single neurons

to solve linearly non-separable tasks [Gidon et al., 2020], while morphologically distinct apical and basal arbors in rodent pyramidal cells support compartmentalized integration during behavior [Otor et al., 2022]. In hippocampal CA1 and CA3 neurons, distal dendrites contribute to place field remapping and flexible memory encoding [Li et al., 2023, OHare et al., 2025], suggesting that dendritic subregions encode functionally distinct computations shaped by morphology and context. More generally, studies such as Beniaguev et al. [2021], de Las Casas et al. [2025] and recent reviews [Tye et al., 2024, Makarov et al., 2023, Payeur et al., 2019] emphasize that nonlinear dendritic subunits enhance representational dimensionality and learning flexibility. Our framework provides a complementary, function-space perspective on these findings, formalizing how dendritic structure constrains the computational regimes accessible to single neurons.

These principles have direct implications for cortical circuit organization. Excitatory neurons with elaborate apical tufts, such as layer 5 pyramidal neurons, may leverage their architecture to flexibly adapt to changing cognitive demands, whereas inhibitory or smaller pyramidal neurons with simpler dendritic arbors promote stability and reliability. Recent work suggests that these structural specializations are coupled to distinct modes of plasticity: in mouse motor cortex, apical and basal compartments follow different learning rules [Wright et al., 2025]. The trade-off between flexibility and robustness we observe echoes these findings. Likewise, our branch-specific weight-freezing experiments parallel dendritic disinhibition mechanisms that gate plasticity and promote contextdependent learning [Canto-Bustos et al., 2022, Onasch et al., 2025]. Modeling efforts that integrate such morphological and compartment-specific constraints could therefore yield more realistic predictions of how cortical circuits balance adaptability with stability.

Network-level implications Beyond the single-cell level, these results have direct implications for network complexity. Neurons with greater intrinsic computational capacity reduce the number of units required to represent a given function, analogous to how richer gate sets affect scaling constants in circuit complexity classes such as NC or AC. Morphological constraints at the single-neuron level could therefore influence the size, efficiency, and modularity of biological circuits. In this view, neuronal diversity does not merely reflect biological variation but an optimization of computational trade-offs; balancing capacity, robustness, and energy cost across scales of organization.

Relevance to NeuroAI Our results also suggest opportunities for artificial architectures. We demonstrate that the same structural trade-offs observed in neurons also arise in artificial architectures. Such dendritic-inspired diversity could help artificial systems balance stability and continual learning. Future work incorporating these hybrid architectures may provide a path toward more adaptive, resource-efficient models bridging biological and artificial intelligence.

Limitations and outlook Our framework simplifies many aspects of neuronal computation. We focus on feedforward architectures with rectified nonlinearities, abstract away spiking dynamics, recurrence, and inhibition, and use Boolean function learning as a proxy for computational complexity. Our analysis is phenomenological; the observed phase transition in realizability has not yet been formally derived from first principles, and its theoretical characterization within circuit complexity or learning theory remains open. Experimental validation, for instance through dendritic imaging or targeted perturbations that test compartment-specific flexibility and robustness, will be essential for linking these abstract results to biological neurons. Nevertheless, the qualitative insights remain robust; morphology directly shapes the computational strategies available to a neuron.

By linking dendritic morphology to single-neuron computational limits via deep network architectures, we provide a tractable, scalable approach bridging cellular neuroscience with theoretical frameworks from machine learning and computational complexity. Future work extending this to ensembles of diverse neurons could reveal how local computational trade-offs influence circuit-level dynamics, shaping cognition, learning, and dysfunction. In short, morphology sculpts computation, shaping not only what neurons do, but how they learn and adapt. Table 1 summarizes how dendritic architectures impose distinct trade-offs between robustness, flexibility, and sensitivity, suggesting that evolution has optimized not just wiring efficiency but computational strategy at the cellular level.

References

Nathan W Gouwens, Staci A Sorensen, Jim Berg, Changkyu Lee, Tim Jarsky, Jonathan Ting, Susan M Sunkin, David Feng, Costas A Anastassiou, Eliza Barkan, et al. Classification of electrophysiological and morphological neuron types in the mouse visual cortex. Nature neuroscience , 22(7):1182-1195, 2019.

Nathan W Gouwens, Staci A Sorensen, Fahimeh Baftizadeh, Agata Budzillo, Brian R Lee, Tim Jarsky, Lauren Alfiler, Katherine Baker, Eliza Barkan, Kyla Berry, et al. Integrated morphoelectric and transcriptomic classification of cortical gabaergic cells. Cell , 183(4):935-953, 2020.

Nikolai C Dembrow, Scott Sawchuk, Rachel Dalley, Ximena Opitz-Araya, Mark Hudson, Cristina Radaelli, Lauren Alfiler, Sarah Walling-Bell, Darren Bertagnolli, Jeff Goldy, et al. Areal specializations in the morpho-electric and transcriptomic properties of primate layer 5 extratelencephalic projection neurons. Cell reports , 43(9), 2024.

Brian E Kalmbach, Rebecca D Hodge, Nikolas L Jorstad, Scott Owen, Rebecca de Frates, Anna Marie Yanny, Rachel Dalley, Matt Mallory, Lucas T Graybuck, Cristina Radaelli, et al. Signature morpho-electric, transcriptomic, and dendritic properties of human layer 5 neocortical pyramidal neurons. Neuron , 109(18):2914-2927, 2021.

Michael London and Michael Häusser. Dendritic computation. Annu. Rev. Neurosci. , 28:503-532, 2005.

Romain Daniel Cazé, Mark Humphries, and Boris Gutkin. Passive dendrites enable single neurons to compute linearly non-separable functions. PLoS computational biology , 9(2):e1002867, 2013.

Brendan A Bicknell and Michael Häusser. A synaptic learning rule for exploiting nonlinear dendritic computation. Neuron , 109(24):4001-4017, 2021.

Albert Gidon, Timothy Adam Zolnik, Pawel Fidzinski, Felix Bolduan, Athanasia Papoutsi, Panayiota Poirazi, Martin Holtkamp, Imre Vida, and Matthew Evan Larkum. Dendritic action potentials and computation in human layer 2/3 cortical neurons. Science , 367(6473):83-87, 2020.

Chris Mingard, Joar Skalse, Guillermo Valle-Pérez, David Martínez-Rubio, Vladimir Mikulik, and Ard A Louis. Neural networks are a priori biased towards boolean functions with low entropy. arXiv preprint arXiv:1909.11522 , 2019.

Claire Kenyon and Samuel Kutin. Sensitivity, block sensitivity, and -block sensitivity of boolean functions. Information and Computation , 189(1):43-53, 2004.

Michael Häusser and Bartlett Mel. Dendrites: bug or feature? Current opinion in neurobiology , 13 (3):372-383, 2003.

Alexandra Tzilivaki, George Kastellakis, and Panayiota Poirazi. Challenging the point neuron dogma: Fs basket cells as 2-stage nonlinear integrators. Nature communications , 10(1):3664, 2019.

Nelson Spruston. Pyramidal neurons: dendritic structure and synaptic integration. Nature Reviews Neuroscience , 9(3):206-221, 2008.

Richard M Karp. Reducibility among combinatorial problems. In 50 Years of Integer Programming 1958-2008: from the Early Years to the State-of-the-Art , pages 219-241. Springer, 2009.

Ingo Wegener. The complexity of Boolean functions . John Wiley & Sons, Inc., 1987.

Panayiota Poirazi and Bartlett W Mel. Impact of active dendrites and structural plasticity on the memory capacity of neural tissue. Neuron , 29(3):779-796, 2001.

Elizabeth Gardner and Bernard Derrida. Optimal storage properties of neural network models. Journal of Physics A: Mathematical and general , 21(1):271, 1988.

David Rubinstein. Sensitivity vs. block sensitivity of boolean functions. Combinatorica , 15(2): 297-299, 1995.

Elchanan Mossel, Ryan O'Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low influences: invariance and optimality. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05) , pages 21-30. IEEE, 2005.

Gil Kalai. Boolean functions: influence, threshold and noise. In European Congress of Mathematics , volume 1, pages 85-110, 2016.

Donald R Jones, Cary D Perttunen, and Bruce E Stuckman. Lipschitzian optimization without the lipschitz constant. Journal of optimization Theory and Applications , 79:157-181, 1993.

Joerg M Gablonsky and Carl T Kelley. A locally-biased form of the direct algorithm. Journal of Global optimization , 21:27-37, 2001.

George Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of control, signals and systems , 2(4):303-314, 1989.

Martha Canto-Bustos, F Kathryn Friason, Constanza Bassi, and Anne-Marie M Oswald. Disinhibitory circuitry gates associative synaptic plasticity in olfactory cortex. Journal of Neuroscience , 42(14):2942-2950, 2022.

Sebastian Onasch, Christoph Miehl, M Maurycy Mi˛ ekus, and Julijana Gjorgjieva. Assembly-based computations through contextual dendritic gating of plasticity. bioRxiv , pages 2025-07, 2025.

Yara Otor, Shay Achvat, Nathan Cermak, Hadas Benisty, Maisan Abboud, Omri Barak, Yitzhak Schiller, Alon Poleg-Polsky, and Jackie Schiller. Dynamic compartmental computations in tuft dendrites of layer 5 neurons during motor behavior. Science , 376(6590):267-275, 2022.

Yiding Li, John J Briguglio, Sandro Romani, and Jeffrey C Magee. Mechanisms of memory storage and retrieval in hippocampal area ca3. biorxiv , pages 2023-05, 2023.

Justin K OHare, Jamie Wang, Margjele D Shala, Franck Polleux, and Attila Losonczy. Distal tuft dendrites predict properties of new hippocampal place fields. Neuron , 2025.

David Beniaguev, Idan Segev, and Michael London. Single cortical neurons as deep artificial neural networks. Neuron , 109(17):2727-2739, 2021.

Eduardo Maristany de Las Casas, Kris Killmann, Moritz Drüke, Lukas Münster, Christian Ebner, Robert Sachdev, Dieter Jaeger, and Matthew E Larkum. Tuft dendrites in frontal motor cortex enable flexible learning. bioRxiv , 2025.

Kay M Tye, Earl K Miller, Felix H Taschbach, Marcus K Benna, Mattia Rigotti, and Stefano Fusi. Mixed selectivity: Cellular computations for complexity. Neuron , 112(14):2289-2303, 2024.

Roman Makarov, Michalis Pagkalos, and Panayiota Poirazi. Dendrites and efficiency: Optimizing performance and resource utilization. Current opinion in neurobiology , 83:102812, 2023.

Alexandre Payeur, Jean-Claude Béïque, and Richard Naud. Classes of dendritic information processing. Current opinion in neurobiology , 58:78-85, 2019.

William J Wright, Nathan G Hedrick, and Takaki Komiyama. Distinct synaptic plasticity rules operate across dendritic compartments in vivo during learning. Science , 388(6744):322-328, 2025.

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: Our work provides a new theoretical framework to analyze the computational properties of single neuron computation. The goal of the paper is to provide proof-ofconcept results showing that different dendritic compartment arrangement can result in differing computational capabilities at the single-neuron level, which we did using toy models.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We have discussed the limitations of our approach in the discussion section, where we talk about the simplifications used in our approach.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [NA]

Justification: Our results are derived from empirical analysis.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: In the main text, we have added modeling details required to reproduce the results. We will make the code available publicly.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: We will provide the GitHub link to the code repository. For reasons of anonymity, we will do so after review.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so No is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: Some details are listed in the main text of the paper, under section 2. Code will be made available to specify finer details.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: We have reported error bars in figures, and how the errors have been calculated.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: We have provided information in section 3. Additionally, we will also share our code through GitHub.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: Yes, we have conformed to the NeurIPS Code of Ethics.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [NA]

Justification: The work does not have direct societal impacts.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: The paper poses no such risks since the models do not generate large-scale datasets.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: We wrote all the code ourselves, and we have cited relevant papers for publicly-available packages.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA]

Justification: No new assets were introduced in the paper.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: Paper does not use crowdsourced data or data from human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: No such risks

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [NA]

Justification: LLMs were not used for the research.

Guidelines:

The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.

Please refer to our LLM policy ( https://neurips.cc/Conferences/2025/LLM ) for what should or should not be described.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the computational and training capabilities of alternatives to standard linear threshold units, to potentially explain the difference in learning ability of standard neural network architectures and human brains. Specifically, it explores two types of replacement units that differ in topology: apical-like units, which are thin and deep, and basal-like units, which are broad and shallow.\n\nTo evaluate the computational complexity of these configurations, the authors analyze the types and proportions of random Boolean functions that each architecture can learn. They focus on functions grouped by similar sensitivity and entropy levels. Their findings suggest that architectural topology significantly influences learning behavior, and that there are non-trivially disjoint sets of learnable functions by each type of neuron.\n\nBasal-like architectures exhibit high computational capacity and show a strong preference for learning functions with high sensitivity. However, they are not very robust and can only be retrained effectively on functions with very low sensitivity. Despite this, they generalize well and tend to learn functions independently, without much feature sharing.\n\nApical-like architectures, in contrast, have lower computational capacity and do not show a strong preference for sensitivity levels. They are more robust, particularly tolerant to variations in initial conditions, but they perform poorly in terms of retrainability and generalization. These architectures tend to share features across functions via their distal branches.\n\nLinear architectures are the simplest of the three, with minimal computational power. They have low sensitivity, poor robustness, and limited retrainability. However, they generalize relatively well. Their computation is purely linear, offering little in terms of complex function learning.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\nThis paper presents an interesting and thoughtful analysis. While the conclusions may not be particularly surprising—given the extensive prior work on architectural variations—it’s valuable to see such a detailed and focused study in the context of Boolean functions. \n\nWeaknesses\n\n1. The notion of entropy used in the paper could benefit from a clearer definition. In the context of Boolean function analysis (e.g., [1]), terms like Fourier entropy have specific meanings that may not align with the usage here. \n\n2. The training procedure should be described in more detail in the main body. Different training setups can lead to different local minima, which may affect the learnability of certain functions. \n\nI also feel the analysis, while interesting, is somewhat simplistic. It would help to know to what extent and in what contexts the conclusions of this could be further generalized and will transfer. A discussion on this will help strengthen the paper. \n\nReference\n[1] Ryan O'Donnell, Analysis of Boolean Functions. arXiv:2105.10386\n\nQUESTIONS:\nIt would strengthen the paper to include a discussion on how the insights from this analysis could inform the design or modification of existing training architectures—particularly in widely used models like transformers for language tasks. For instance, if certain topologies (e.g., basal-like) are shown to be more robust or generalizable for specific classes of Boolean functions, could similar structural biases be introduced into transformer blocks to improve their inductive biases or training dynamics?\n\nMore concretely, it would be helpful to know:\n\n1. Whether these findings suggest new ways to structure attention heads or feedforward layers. \n2. If certain types of functions (e.g., high-sensitivity ones) are better captured by specific architectural motifs. \n3. How these insights might guide the design of hybrid models or inform pruning strategies.\n\nA forward-looking discussion on these implications would make the work more actionable and relevant to the broader machine learning community.\n\nLIMITATIONS:\nYes, but it might be better to have a separate \"Limitations\" section in the paper.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-17"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper investigates how the dendritic morphology of single neurons constrains their computational capacity, using abstract neural network models inspired by biological dendritic architectures. The work discusses the computational capability of two types of dendritic morphology by using three types of Boolean functions. \nThe authors discover a phase transition in functional realizability with increasing input dimensionality. They show that this transition is strongly influenced by the dendritic architecture and the number of nonlinear compartments.\nThese findings suggest that morphological features of neurons impose structural inductive biases that shape their computational strategies. The work opens up new directions for incorporating biologically grounded neuron models into machine learning, particularly for tasks involving continual adaptation and resource efficiency.\nThey reveal also that “basal” architectures are more robust and perform well on low-complexity functions. “apical” architectures are more flexible, better at generalization and retraining, especially after pretraining on simple tasks. These findings suggest that morphological features of neurons impose structural inductive biases that shape their computational strategies. The work opens up new directions for incorporating biologically grounded neuron models into machine learning, particularly for tasks involving continual adaptation and resource efficiency.\n\nSTRENGTHS AND WEAKNESSES:\nThe authors innovatively employ three classes of Boolean functions to evaluate the computational capabilities of different dendritic neuron models (apical and basal), and offer a compelling discussion on the robustness and flexibility trade-offs between architectures. However, several concerns remain:\n1. The modeling details are insufficiently specified, making it difficult to assess the fairness and reproducibility of the architectural comparisons.\n2. The description of the Boolean functions is vague, and it is unclear how representative or diverse these functions are.\n3. While the discovery of a critical dimension associated with a phase transition is intriguing, the experiments are conducted on a relatively small function set, and the lack of accompanying theoretical analysis limits the generality of the findings.\n4. Although the authors provide empirical support for their claims, the scalability and practical relevance of the proposed approach remain uncertain due to the restricted experimental scope.\n\nQUESTIONS:\n1. The authors present abstract models for the linear, apical, and basal architectures. However, the specific structural differences among these models are insufficiently described, making it difficult to align the experimental results with the stated conclusions. For example, the “linear” variant appears to be a degenerate form of the basal model without activation functions, yet this distinction is not clearly articulated. Furthermore, to ensure a fair comparison, the number of parameters in the linear model should be matched with those in the apical and basal models.\n2. While the authors explore the impact of dendritic morphology on Boolean functions across input dimensions ranging from 3 to 8, this dimensional range is quite limited compared to real-world applications. It remains unclear how the findings scale to higher-dimensional inputs, which are more representative of biological and artificial neural systems.\n3. The authors introduce three types of Boolean functions, i.e., typical, fixed entropy, and fixed sensitivity, but do not clearly explain how these function sets are used to evaluate model performance, nor how the ratios or distributions of these sets are determined. This obscures the interpretability and reproducibility of the experimental setup.\n4. The paper reports an interesting phase transition in function realizability at a certain critical input dimension, which appears to correspond roughly to the number of dendritic branches. However, it is unclear whether this critical dimension can be formally predicted or calculated based on model parameters. Providing a theoretical estimate or analytical expression for this threshold would significantly strengthen the paper’s contributions.\n5. In Section 5, the authors investigate retraining dynamics by freezing the weights of specific dendritic branches. However, the branch indexing scheme is not clearly defined, making it difficult to interpret which components are being constrained. This lack of clarity weakens the conclusions regarding retraining flexibility across architectures.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-18"}

comments: SUMMARY:
This work explores how dendritic morphology constrains the computational capacity of individual neurons by modeling them as feedforward networks with non-linear dendritic compartments. The authors compare basal-like and apical like morphology vs point neuron architecture to  analyze which classes of Boolean functions are realizable by each. The study reveals a sharp phase transition in function learnability with increasing input dimensionality, provides insights into architecture-specific trade-offs between robustness and flexibility, and connects findings to implications in neuroscience and artificial intelligence.

STRENGTHS AND WEAKNESSES:
Strength:
The work systemtically inspects function realizability across dimensions and complexities. This can help us gain better understanding the role played by dendritic architecture.
The distinction between apical and basal architectures is a important contribution.

Weakness:
The assumption of data input dimensionality is very far from reality (N_dim ≤ 8). The study assume a single neuron need to be responsible for seperating input patterns invidivually. In reality, neurons only receives very spare (and very high dimesion) inputs. The information is likely encoded in superposition manner. Likely the learning rely on the coordination of many neurons.

QUESTIONS:
While this simplified analysis can help us gain insight on the underlying mechanism that might offer advantage for dendritic architecture.  I suggest authors to show that their theory is still relevant under realisitc datasets and model of more than a single neuron.

LIMITATIONS:
No. Please see Strengths And Weaknesses.
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper computationally examines how a simplified model of neuron morphology can impact the types of functions it can learn. The authors show that \"basal\" like neurons learn a range of functions, however are not able to generalize well, while \"apical\" like neurons can quickly generalize, but learn less reliably. These results provide insight into why the brain has both apical and basal dendrites.\n\nSTRENGTHS AND WEAKNESSES:\n**STRENGTHS**\n\n1. This paper is well motivated with a clear Introduction and Abstract. \n\n2. Figure 1 is very clear and really helps the reader understand the methods of the paper. \n\n3. The paper was well scoped - it took on just the right amount enabling a clear result that will be of interest to the NeurIPS community. \n\n4. The re-training results are quite interesting and suggest and interesting trade-off. \n\n5. I love the summary \"morphology sculpts computation, shaping not only what neurons do, but how they learn and adapt\". I think that is a really great conclusion of the authors work (and could possibly even be added to the Abstract). Table 1 is also a helpful summary. \n\n6. The implications for deep learning are interesting and I enjoyed that they were commented on.\n\n\n**WEAKNESSES** \n\n1. The first 3 sections of the paper were (largely) very clear and well written. The main weakness of this paper is that the next sections feel a little more rushed and less clear, particularly Section 5. I think there are a few straightforward things that could be done to address this: \n    a. While I appreciate that Fig. 3C illustrates the authors' point about the difference between apical and basal dendrites based on how well they learn functions of different sensitivities, I feel like there might be a better way to show this. Possibly is matrix heat map, where each square is the mean sensitivity for each of the different pairwise probability values (I know that these values varied, but the plot currently already has a block like structure, so taking the average across all the dots in the block and plotting in a heat map would make it easier to read). \n    b. Fig. 4 should be made bigger. It's hard to read as is and I think makes it much harder to grasp the authors' point about the difference between the neuron types and their re-training abilities. This could be achieved by moving the legend of Fig. 4C underneath the plot (instead of to the side) so that the figure can be scaled larger. \n    c. I think the Discussion section (which is well written) could be strengthened by more citations/connection to existing literature (especially the neuroscience literature). \n\n2. This work cites relatively few other papers. While I understand the preference of not citing things that are not essential to the authors' work, I think the lack of citations makes it harder for the reader to understand exactly how this work situates itself within the existing literature. Especially commenting on more neuroscience literature would be helpful - there is work studying basal and apical dendrites and their computations. How does this work compare? How might a neuroscience experiment test the authors' conclusions? What implications does this work have for studies showing that basal and apical dendrites in hippocampus are not stable? (These are just a few thoughts that came to mind - no need to address any or all of them, but thinking some more about some connection I think would be good). I was also surprised Beniaguev et al. (Cell 2021) was not cited in the Introduction when discussing the computational roles of dendrites. It's my understanding that that is a landmark paper that has shaped how people in the community think about the computational effect of dendrites. \n\n**MINOR POINTS**\n\n1. I think this work would benefit from a careful re-read to catch typos (there are several instances where there are no spaces between words) and inconsistencies in how equations and figures are referenced. For instance, Figure 1 a, Figure 1B, Figure 1 B, Figure2B, Supplemental figure 6, and 1(a) are all used to refer to specific figures. Similarly, Equation 7 and 5 are used to refer to specific equations. Making it more consistent would help the reader (although, I recognize this is a more minor point).\n\nQUESTIONS:\nI do not have any questions for the authors.\n\nLIMITATIONS:\nThe authors nicely addressed their limitations in the Discussion section.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the computational ability of single neurons, which are modeled as simple feedforward relu networks. Both deep/narrow and wide/broad network topologies are considered for the neurons dendrites, each of which have different pros and cons, highlighting that the morphology of neurons leads to different functionality at the level of single neurons.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* Individual neuron complexity, and how it effects computation in neural networks, is theoretically understudied and an interesting research direction\n* The problem motivation and approach are clearly written and easy to understand.\n* The figures are mostly clear and easy to read\n* Limitations are clearly laid out, and leave a lot of open directions\n\nWeaknesses:\n* The biggest weakness is in the clarity and possible robustness of the training (see questions below)\n* They discuss effects of neuron morphology on the level of single neuron computation, but it is not clear in what way these morphologies impact computation at the level of networks of many neurons. Making stronger connections between single cell computational ability and the capabilities of networks composed of these neurons would strengthen this work.\n\nQUESTIONS:\nThe biggest question I had was about the training protocol used, which is not clearly outlined either in the main text or supplemental. I would recommend replacing comments about training times with clear outlines on how these networks are trained. This is especially important here, as it is not obvious from the text if the network fails to learn a particular boolean function because the network architecture does not support it, or if it is because of a failure due to the learning protocol (due to, say, a nonconvex learning landscape). I believe this is why the authors also include results for best case realizability, it would be good to make this clear in the text. Even with multiple learning trials for a single function, there are situations where the learning protocol may never be able to realize a function that is still possible to implement by the network. I think it would be good to make this limitation a bit more clear.\n\nThis paper also effectively targets questions about functional realizability in terms of network breadth and depth. There is a bit of literature on this topic, see for example M. Telgarsky (2016) or D. Rolnick & M. Tegmark (2018). Though I'm not sure if these results are immediately applicable, they might be a good direction to look into.\n\nSome smaller nitpicks:\n* What is being shown in figure 3B could be made more clear in the description\n* The text in figure 4 is very hard to make out\n* Although a bit beyond the scope of this paper, even some cursory remarks on the effect of local, neuronal level, functional realizability on the behaviour of larger networks I think would make this paper more interesting to the larger NeurIps community.\n\nNot so much a question for this paper, but there a lot of interesting directions to take this type of study. \nAlong with the many directions listed in the discussion, it would be interesting to explore how this extends to neurons with multiplicative presynaptic interactions, such as those that can occur during presynaptic gating, as such interactions are fundamentally different then the additive ones used in standard ANNs but quite relevant for biologically plausible organizations\n\nLIMITATIONS:\nDiscussed in strengths and weaknesses and questions", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}