How Biological Synapses Self-Assemble Gradient Learning

Qianli Liao; Liu Ziyin; Yulu Gan; Brian Cheung; Mark T. Harnett; Tomaso Poggio

cs.NEDec 28, 2024

How Biological Synapses Self-Assemble Gradient Learning

Authors: Qianli Liao, Liu Ziyin, Yulu Gan, Brian Cheung, Mark T. Harnett, Tomaso Poggio

Organizations: Center for Brains, Minds, and Machines, MIT · McGovern Institute, MIT · Research Laboratory of Electronics, MIT · Physics & Informatics Laboratories, NTT Research ·

Abstract

Existing models of learning in the brain explain how given circuits learn, but not how biology could assemble those circuits in the first place. To address this gap, we formulate Self-Assembling Learning (SAL)---the study of how learning systems can emerge from lower-level interactions---and introduce one example mechanism, the Self-Assembling Motif (SAM). SAM is self-assembling at two scales: the motif emerges from initially random connectivity under heterosynaptic plasticity rules, and networks of SAMs, composed hierarchically, self-organize into dynamics that provably approximate a generalized form of stochastic gradient descent---matching backpropagation-level performance. This suggests that biological learning need not be prescribed but can emerge from local rules---and to a far greater extent than previously thought.

Figures & tables

Figure 1: A model is biologically plausible only if biology could produce it on its own: biological plausibility/realizability as a graded axis of self-organization. Learning algorithms differ in how much of their structure self-assembles from local interactions rather than being stipulated in advance. The vertical axis is an ordinal ladder of organizational levels: the inference/forward weights, the circuit that adjusts them, the scaffold that hosts it, and beyond. We label the second level the credit-assignment circuit to be concrete, though more generally it is the learning circuit —the machinery that turns experience into inference weight changes, of which credit assignment is the canonical example. Backprop and feedback-alignment variants self-organize essentially only their weights, leaving this circuit fixed; SAM assembles the circuit itself from uniform rules, resting only on a minimal two-stream scaffold whose form is largely required for stable perception. The faded top indicates that the axis is open-ended.

Figure 2: A synaptic motif for interactions between ascending and descending information streams. (a) Schematic of the upstream-downstream architecture. The upstream consists of a standard fully-connected neural network with multiple layers, roughly corresponding to a multi-region cortical processing pathway (e.g. V1-V2-V4-IT). The output of the upstream network goes to an error processing module (potentially corresponding to frontal cortices). This module computes a local error signal. This error signal is sent to the feedback (downstream) pathway, which processes information layer by layer downwards. The orange arrows represent non-learnable (identity) connections. Blue arrows represent learnable connections, each parameterized by a fully-connected weight matrix. (b) A detailed view of the smallest unit of the connection motif in panel a, indicated by grey dashed box. The neurons in panel b (as well as the abstract forms in A and C are illustrated as pyramidal neurons, the principal cell type of the cortex. Only apical dendrites are illustrated here for simplicity. (c) A mathematical description of the unit in panel b. Each arrow (which corresponds to a collection of axons, dendrites, and synapses) represents a set of full connections between two groups of neurons, parameterized by a weight matrix W or V . Every h in C is a vector and refers to the activations of a group of neurons. The connection matrix V allows upstream and downstream networks to have different number of hidden units in corresponding layers.

Figure 3: Detailed analyses, variants and ablation studies of SAM. (a) The matrices $\bar{V}\bar{V}^{T}$ (upper) and $WW^{T}$ (lower) before and after training. Also, recall that $\bar{V}\bar{V}^{T}=H^{\ell}$ determines the ideal aligned matrix learning rate $H_{D}^{\ell}$ for $W^{\ell}$ at $E_{v}^{\ell+1}=0$ (Theorem 1 ; more generally the prefactor is $H_{D}^{\ell}+\widetilde{E}_{v,D}^{\ell}$ ). The neurons within the same layer become correlated after training. The spectra of the weight matrices of the two pathways are found to be effectively low-rank (right). This visualization used the SVHN dataset with a network having 8 hidden layers. (b) Ablation study on roles of $\bar{V}$ , $\bar{W}$ and $V$ on CIFAR-10. Here, we make a subset of the interconnections nontrainable. The legend labels indicate trainable components. “Linear” refers to a linear network trained with SGD. We see that (1) making $\bar{V}$ plastic is more important than making other connections plastic, and (2) making everything plastic significantly improves the performance further. (c) The impact of the width of the downstream pathway on performance. We use different activation functions in the forward pathway. The width of the upstream pathway was kept constant at 200, while varying the width of the downstream pathway. We find that having a wider downstream network improves the performance of the feedforward network up to an overparametrization ratio of 7.5. In biology, the ratios can also depend on biological constraints such as energy consumption or wiring volume and may be region-dependent.

Figure 4: Alignment diagnostics on CIFAR-10 (3 hidden layers; stable run). SAL vs. backprop during a

50

-epoch run (hyperparameters in SI Alignment Diagnostics). A: train and test cross-entropy and accuracy over epochs. B: cosine similarities between SAL and backprop gradients on

W^{\ell}

increase during learning and reach high level. C–D: pathway error

\|E_{p}^{\ell}\|

and

\|E_{p}^{\ell}\|/\|\mathcal{F}^{\ell}\|

(

E_{p}^{\ell}=\mathcal{F}^{\ell}-\mathcal{B}^{\ell}

, normalized by

\|\mathcal{F}^{\ell}\|

); both decrease on most layers;

E_{p}

tracks pathway misalignment and dominates the residual once

E_{v}

is small. E–F: interstream gradient magnitudes

|\nabla V^{\ell}|

and

|\nabla\bar{V}^{\ell}|

remain at a stable positive level. G–H: interstream mismatch

|V^{\ell+1}-\bar{V}^{\ell\mathrm{T}}|

falls and junction cosine similarity approaches one, consistent with unconditional

E_{v}^{\ell}\to 0

even while interstream plasticity stays active. Together, tight interstream alignment and decreasing pathway error match the conditional main theorem and help explain why SAL gradients track backprop (panel B). These are early diagnostics; a systematic analysis of

E_{p}

is left to future work.

Acc. $\uparrow$	CIFAR10	MNIST	Fashion MNIST	ChestMNIST	PathMNIST	SVHN	STL10
SGD	50.36 $\pm$ 0.16	98.44 $\pm$ 0.09	88.63 $\pm$ 0.26	73.26 $\pm$ 0.14	70.11 $\pm$ 0.10	78.20 $\pm$ 0.13	41.00 $\pm$ 0.07
FA	47.85 $\pm$ 0.20	97.80 $\pm$ 0.19	85.41 $\pm$ 0.26	70.44 $\pm$ 0.27	69.90 $\pm$ 0.10	78.03 $\pm$ 0.15	39.29 $\pm$ 0.23
WM	49.35 $\pm$ 0.13	98.40 $\pm$ 0.23	88.34 $\pm$ 0.36	70.14 $\pm$ 0.08	70.00 $\pm$ 0.15	78.08 $\pm$ 0.15	39.48 $\pm$ 0.19
KP Algorithm	48.16 $\pm$ 0.09	98.45 $\pm$ 0.09	88.52 $\pm$ 0.25	71.03 $\pm$ 0.10	69.51 $\pm$ 0.05	78.20 $\pm$ 0.05	40.32 $\pm$ 0.11
SAM (ours)	51.91 $\pm$ 0.21	98.85 $\pm$ 0.10	88.86 $\pm$ 0.17	73.21 $\pm$ 0.11	70.23 $\pm$ 0.15	78.50 $\pm$ 0.07	41.50 $\pm$ 0.10

Table 1: Performance of the proposed method compared with SGD and biologically plausible algorithm baselines. Bold denotes the best performing algorithm, and italics denote the best runner-up algorithm. We see SGD and SAM are comparable, while significantly outperforming existing biologically plausible learning algorithms. For each experiment, we conduct five runs and calculate the error bars.

	Connectivity				Plasticity
	$u\to u$	$d\to d$	$u\to d$	$d\to u$	$u\to u$	$d\to d$	$u\to d$	$d\to u$
SGD	✓	✗	✗	✗	✓	✗	✗	✗
Feedback Alignment	✓	✓	✗	✗	✓	✗	✗	✗
Weight Mirroring	✓	✓	✗	✗	✓	✓	✗	✗
SAM	✓	✓	✓	✓	✓	✓	✓	✓

Table 2: The upstream-downstream table of biologically plausible learning , emphasizing the necessity for four types of connections between the two pathways: downstream to downstream (

d\to d

), downstream to upstream (

d\to u

), upstream to downstream (

u\to d

), and upstream to upstream (

u\to u

). Each connection must exhibit plasticity. For a learning rule to be biologically plausible, it must demonstrate both connectivity and plasticity.

SAM pathway	Weight type	Cortical pathway example	Dendritic targeting	Observed influence type
u $\to$ u	$W$	L4 $\to$ L2/3 (within area)	Basal / proximal	Driving
		L2/3 $\to$ L4 (higher area)	Basal / proximal	Driving
u $\to$ d	$V$	L2/3 $\to$ L5 (within area)	Basal $\,+$ apical	Conditionally driving
d $\to$ d	$\bar{W}$	L5 $\to$ (thalamus $\to$ ) L5 (inter-areal)	Basal $\,+$ apical	Conditionally driving
		L6 $\to$ (thalamus $\to$ ) L6 (inter-areal)	Basal / proximal	Driving
d $\to$ u	$\bar{V}$	L5 $\to$ L1 $\to$ L2/3 (inter-areal)	Apical / distal	Modulatory

Table 3: One possible correspondence between canonical cortical pathways and SAM pathway types. Dendritic targeting and observed influence type describe biological properties of the cortical implementation rather than assumptions of the SAL/SAM framework. “Conditionally driving” refers to a biophysical property observed in some layer 5 pyramidal neurons, arising from interactions between basal and apical compartments, and is not a requirement of SAM itself.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Effective learning circuits self-assemble from a randomly initialized pool of neurons under the heterosynaptic rule, with no two-stream scaffold. Reproduced from [ 50 ] . (a) Examples of evolved four-neuron circuits trained on a linear regression task; synapses are grown and pruned over generations. (b) Snapshots of an evolving $100$ -neuron network. (c) Quantities versus generation (rows: four-neuron sparse init., $100$ -neuron sparse init., $100$ -neuron dense init.), showing fitness, gradient alignment, heterosynaptic (HSP) density, Hebbian (HBP) density, and synapse density; shaded regions indicate variability across runs. As evolution proceeds, the evolved updates become increasingly aligned with the true gradient.

Figure 6: Performance of a four-layer MLP for different choices of learning rates. Interestingly, the best performance is achieved when $\eta_{V}$ is the smallest of the four learning rates.

Figure 7: Alignment diagnostics on CIFAR-10 (3 hidden layers,

L=4

; stable run). SAL vs. backprop comparison during training (hyperparameters above). A: train and test cross-entropy and accuracy over epochs. B: cosine similarity between SAL and backprop gradients on

W^{\ell}

. C–D: pathway error

\|E_{p}^{\ell}\|

and

\|E_{p}^{\ell}\|/\|\mathcal{F}^{\ell}\|

(

E_{p}^{\ell}=\mathcal{F}^{\ell}-\mathcal{B}^{\ell}

, normalized by

\|\mathcal{F}^{\ell}\|

; typically decreasing). E–F: interstream gradient magnitudes

|\nabla V^{\ell}|

and

|\nabla\bar{V}^{\ell}|

. G–H: interstream mismatch

|V^{\ell+1}-\bar{V}^{\ell\mathrm{T}}|

and cosine similarity. Interstream self-assembly completes while plasticity remains active; pathway error falls on most layers; SAL tracks backprop on

W^{\ell}

. These are early diagnostics; a systematic analysis of

E_{p}

is left to future work.

Figure 8: Alignment diagnostics on CIFAR-10 (4 hidden layers,

L=5

; deliberately chosen divergent run). Same panel layout and default hyperparameters as Figure 7 , with four hidden layers; divergence is not inevitable at this depth and depends on learning-rate tuning.

\|E_{p}^{\ell}\|

can decrease on some layers early, while rising around the time instability appears; interstream alignment in panels G–H can remain good on all layers when failure happens.

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The brain learns abstract representations of high-dimensional sensory input, but the plasticity rules that enable such learning are unknown. We study biologically plausible algorithms on the Random Hierarchy Model (RHM), an artificial dataset designed to investigate how deep neural networks learn the intrinsic hierarchical structure of high-dimensional data. We focus on two types of local learning rules that avoid both a long convergence time and the use of a symmetric error network. The first type uses direct feedback signals to approximate error propagation from the output layer. The second type uses layerwise self-supervised contrastive or non-contrastive loss functions that do not explicitly approximate errors at the output layer. We show that all rules of the first type fail to solve the tasks of the RHM and trace this failure back to input-specific nonlinearities (`masking') that are implemented in full backpropagation and are essential for learning complex tasks. However, algorithms of the second type are able to learn the hierarchical hidden structure of the RHM tasks and are as data-efficient as supervised backpropagation training, while being compatible with known rules of synaptic plasticity in cortex.

Abstract

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