1.1. Background

The error backpropagation algorithm is the state of the art in training artificial neural networks (ANNs), with immense advances in practical applications made over the past decade. On the other hand, neuroscience has revealed great functional diversity across brain areas and an entire zoo of cell types. However, models taking into account such biological constraints have only produced limited understanding of how microscopic learning leads to complex function. Adapting individual components in a very complex network requires a meaningful local representation of errors; this has sparked the ‘backpropagation and the brain’ hypothesis [1,2], i.e., that biological neural networks learn via a form of the error backpropagation algorithm (BP) powering deep learning.

However, the biological implausibility of BP has been a long-standing point of objection [1,3]. Why would one then still consider backpropagation as a viable candidate for explaining learning in the brain? We argue that this is a consequence of applying Occam’s razor to physiological evidence: Mismatch responses (in other words: error encoding) have been observed in a variety of brain areas and across species, such as humans [4], non-human primates [5,6], mice [7–11], and birds [12]. In particular, experimental findings suggest that cortical layer 2/3 (L2/3) pyramidal cells (PYR) function as error coding neurons, with evidence across many cortical areas, e.g., visual areas [4,13–16], visuomotor [17], motor [18], auditory [19]. Error neurons have also shown to emerge naturally in simulated RNNs trained under constraints of energy consumption [20]. If cortex has explicit error units, and all neurons are plastic, then it should arguably use such errors to their fullest.

Currently, BP appears to be the most efficient way for training multi-area networks in terms of generalizability, scalability, and task complexity. This is because BP propagates local errors across all areas, tailored to each neuron according to its contribution to task performance (credit assignment). Since BP works so well on artificial networks, our aim is thus to discover whether the cortical hierarchy could implement a bio-plausible variant of the BP algorithm. In this work, we suggest a concrete circuit implementation, which provides experimentally testable predictions and performs well on many different benchmarks, and compare it to alternative suggestions.

1.2. Related work

A contentious point of discussion is the (in-)compatibility of BP with the predictive processing framework (predictive coding, PC). However, as highlighted by [21–25], there is a fundamental relationship between the computations performed in hierarchical predictive coding (hPC) and ANNs trained with error backpropagation. This relationship is well summarized by two properties. First, the structure of error propagation in hPC at equilibrium of neuronal dynamics is the same as the one of backpropagation, as first remarked by [21]. Second, the representations in hPC after neuronal inference and in BP after a forward pass are not necessarily the same, such that ‘vanilla’ hPC and BP do not lead to the exact same weight updates. This motivates a ‘fixed prediction assumption’, making the equivalence strict by using the activity induced by a forward pass in the learning rule of hPC. Multiple authors have also remarked that peculiar choices of hyperparameters (short inference with Euler-dt 1) also makes the equivalence strict [24,26]. Crucially, this means that the ‘backpropagation and the brain’ hypothesis and predictive coding provide similar solutions to the credit assignment problem. They can be seen as closely related learning algorithms, which have traditionally been employed on different task domains: predictive coding networks (PCNs) are commonly trained to predict stimuli, which is a generative task, while ANNs historically arose as solutions to classification problems. It is important to realize that the task domains are related by a simple exchange of stimuli and latent, and an inversion of the network hierarchy. Our model is built on this insight, and able to reflect it in its connectivity, with either a generative or classifier configuration (see Section 2).

While PCNs can be successfully trained on various tasks and scale to many areas [27,28], most studies do not detail how biology can implement the required computations. For example, the classical PC model by Rao and Ballard [29] is limited in its biological plausibility due to issues of ideal signal transport between neurons, weight copying, strictly hierarchical connectivity, and neuronal non-linearities (see Section 4.2).

Recent work aims to address these issues one by one. Two models stand out in particular due to their biological plausibility: the dendritic cortical microcircuit model of Sacramento et al. [30], and the dendritic hPC model by Mikulasch et al. [25], both of which propose that errors are constructed on dendrites of pyramidal cells via an interplay of local excitation and inhibition. This idea fits well into the theory of balanced networks and predictive coding, but assumes local interneurons exactly copying pyramidal neuron activity, which is unlikely given that interneurons integrate inputs from different cells, sometimes even non-linearly [31]. Moreover, models with dendritic error construction do not scale to more than two areas, as we will demonstrate using extensive simulations.

In contrast, we propose a model using recurrently connected prediction and error streams, loosely implementing the forward and backward pathways of error backpropagation. Similar connectivity is explored in attention-gated reinforcement learning (AGREL) and variants [32–36], but without solutions for the above-mentioned issues of bio-plausibility.

The most important computational achievement of our model is that it scales to many areas, while maintaining all of its bio-plausible constraints. Many studies in the literature have significantly simplified their model dynamics when scaling up simulations, e.g., by assuming ideal weight sharing [25,30], replacing neuron dynamics by steady-state approximations or artificially re-initializing voltages [30,37,38], or implicitly removing recurrency by calculating separate forward and backward passes [30,39]. Here, we will demonstrate that our model is able to scale without changing the implementation of its dynamics in any way.

1.3. Summary of our model

Our model addresses a large number of biological constraints, while establishing a clear relationship to BP. We model two distinct subpopulations of cortical pyramidal cells in each area: one error and one representation (= prediction) population, corresponding to L2/3 and L5 neurons respectively. Connections between error and representation units in different areas are based on the functional connectome of the primate visual cortex [40] (see Fig 1 b1). Thus, inter-area connectivity does not need to be strictly hierarchical, as opposed to most PC models in the literature. Furthermore, there is no biologically implausible one-to-one matching of prediction and error units within each area.

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Fig 1. Neuron and network model.

a1) Morphology: our model neurons have three compartments (soma, error and representation), which may be realized by various morphologies with apical, basal or perisomatic dendritic integration zones. a2) Dynamics: two types of pyramidal cells encode representations and errors, using distinct dynamics. a3) Connectivity: Between areas (left), two opposing streams of representation and error signals form, with preferential and weaker projections (opaque vs. semi-transparent arrows, respectively). Locally (right), representation units receive inputs from other areas onto their prediction-receiving compartment, and local error information on the error-receiving compartment. Error units integrate local representations and errors projected from other areas in their two dendritic compartments. b1) Projection densities across the Macaque visual cortex, used for the inter-area connectivity in our model. Reproduced from Markov et al. 2014 [40]. b2) Generative and classifier configurations of the model differ by inputs, targets, and area labels. In the (supervised) predictive coding configuration, the model receives a latent representation (‘input’) and learns to predict bottom-up stimuli (‘target’). Errors are projected downstream in the hierarchy of cortical areas. The ANN configuration has input and target exchanged, the task becoming digit classification. The information flow is inverted, and representations are projected downstream.

https://doi.org/10.1371/journal.pcbi.1014164.g001

In order to integrate both predictions and errors, each neuron is modeled with three separate integration zones (‘compartments’): one somatic and two separate dendritic compartments. For example, neocortical pyramidal cells are characteristically defined by their somatic, basal and apical dendritic integration zones [41,42]. More generally, our model can be adapted to different neuronal morphologies, see Fig 1 a1. Neuron dynamics are implemented in continuous time, without discrete phases of information transfer (i.e., no distinct forward and backward passes as in ANNs and some bio-plausible models [30,32–35]). The dynamics of each neuron type, in conjunction with its connectivity, encode their function as either error or representation units. Error units make use of non-linear dendritic dynamics, and communicate locally within each cortical column, providing a neuron-specific learning signal to each representation unit. This is important, because it treats error units as distinct cells with their own dynamics, without requiring knowledge of the somatic potential of local representation units (see Section 4.2). All neurons implement prospective rate coding [43,44], modeling the diverse temporal signatures of neuronal outputs, and providing short-term memory to solve temporal tasks.

Learning is performed via a local Hebbian-type rule of the form presynaptic rate × postsynaptic voltage difference (delta rule). Similar to the voltage dynamics, learning is phaseless and always-on, as opposed to the vast majority of bio-plausible theories, such as PCNs [21–24,29], contrastive Hebbian learning [45,46], the wake-sleep algorithm [47], difference target propagation and variants [48–50], AGREL [33,51], and equilibrium propagation [37].

In this work, we will demonstrate the capabilities of our model to solve complex tasks and scale training to many cortical areas on par with ANNs, and compare in particular to the models of dendritic error construction [25,30]. Compared to existing bio-plausible theories of learning, the main novelty of this work is that it includes all of the following:

• successful learning without strict hierarchy of areas (→ inter-area connectivity);
• modeling of PYR as multi-compartment neurons with temporal dynamics (→ morphology ff.);
• phase-less, always-on learning (→ plasticity);
• no strict one-to-one matching of representation and error units (→ plasticity);
• efficient learning across many areas (→ simulation results).

2.1. Inter-area connectivity

Like ANNs, PCNs derive computational efficiency from hierarchical implementations; however, biological networks are never strictly hierarchical. To accurately model the degree of hierarchy found in cortex, we base the inter-area connectivity of our network on the projection densities found among the first areas of the Macaque visual cortex [40]. There, one may describe a rough order between areas defined by the anatomical hierarchy of visual cortex [40,52,53] and the anterior temporal lobe [54]; however, functional connectivity shows projections also between non-neighboring areas (Fig 1 b1). For our model, this means that instead of a strict hierarchy, connections between all areas are allowed (the equivalent of skip connections in ANNs). This ‘loose hierarchy’ will allow us to compare learning performance to ANNs and strictly hierarchical PCN [25].

The network admits two configurations in terms of this ordering (Fig 1 b2): one where errors are passed from early to downstream areas (‘generative’ or ‘PC’ setup), and an inverted order, where representations are passed downstream (‘classifier’ setup). In the generative configuration, the task of representation neurons is to predict activity in primary areas, as in the Rao-Ballard model [29], albeit with different connectivity and neuronal dynamics (see Section 4.2). (We differentiate between the principle of predictive coding, i.e., a generative model where representations are projected top-down, and the specific predictive coding model by Rao and Ballard.)

In this section, we keep the discussion of our model general to the configuration, and refer to the streams as representation (green in Fig 1 a3) and error streams (red) (note that in the context of predictive coding, representation neurons are also known as prediction units). Our model is thus quite general and can encompass various hypotheses of cortical architecture, such as different directions of information flow or projection densities between and within areas.

Stimuli and latent activations are presented to the ‘input’ and ‘target’ areas, which are defined depending on the configuration (Fig 1 b2). For example, for the generative configuration, visual stimuli are presented to V1 as targets, while latent priors are provided to the downstream area V4 (‘input’); for sharp priors, e.g., in the form of labels, this amounts to supervised training. By modeling different latent priors, our model can also adapt other learning schemes, such as unsupervised learning as in the Rao-Ballard model (see Section 4.4).

2.2. Local neuron populations

In each area, we model two populations of pyramidal cells, acting as representation and error units (Fig 1 a2). These populations may be realized by cortical L2/3 PYR, corresponding to the error neuron populations (red in Fig 1), and L5 PYR, corresponding to the representation units (green). Across areas, information is passed between representation units through uni-directional synapses W, while errors are calculated at the ‘target’ area, and projected between error units through uni-directional weights B (Fig 1 a3, right). Within each area, representation and error units are connected via local weights L^ER (representation → error) and L^RE (error → representation), see Fig 1 a3.

We model each pyramidal neuron with three dendritic integration zones: a somatic, a representation and an error compartment. In real PYR, these compartments may be realized by a range of different morphologies, composed of, e.g., somatic, basal, perisomatic and apical integration zones. Depending on the type of neuron (representation or error unit), these compartments perform different calculations and observe different connectivity (Fig 1 a2), as we describe in the following.

2.3. Representation units: morphology

L5 PYR acting as representation units integrate all of their inputs in the soma, as in the classical neuron model of Petreanu et al. [55]. Importantly, to provide the correct error signal to a synapse W located at a representation-receiving dendrite, our model assumes that the somata additively integrate afferent error and representation signals (see neuron dynamics, below), with negligible non-linear distortions.

As illustrated in Fig 1 a1, such additive integration can be realized in biology by different morphologies. If both errors and representations are coded in perisomatic integration zones (Fig 1 a1, 1st morphology), the soma is able to sum the inputs with minimal non-linear effects.

However, when considering morphologies with long apical dendrites typical to L5 PYR (2nd & 3rd morphology), PSP dispersion and non-linear attenuation of somatic activity need to be taken into account. For example, if the representation neuron’s error-receiving compartment consists of a basal integration zone, and predictions are received in apical tufts (2nd morphology), the error signal provided to the synaptic site W will be attenuated after propagation along the apical dendrite. Conversely, error- and representation-receiving compartments may be swapped (3rd morphology), as hypothesized also in [30,56]. If error units are L2/3 PYR, this means that L5 PYR representation units receive local error signals through apical dendrites reaching L1, in line with experimental data [19,57]. However, in this morphology, apical errors are subject to dispersion when propagating to the soma. Furthermore, experiments have shown that top-down or local L2/3 input to L5 PYR non-linearly amplifies somatic activity [58,59], e.g., by triggering Ca²⁺-mediated plateau potentials.

In an alternative realization of our model, errors and representations are both received at the apical tuft. In this scenario, the additive integration could be performed by the apical tuft, and the sum would be propagated to the soma. This agrees with findings that the somata of L5 PYR are largely driven by apical tuft activity [42,58], and fits the results of Francioni et al. [56].

2.4. Representation units: neuron dynamics

Representation units integrate two streams of information: 1) representation units projecting from other areas onto a prediction-receiving compartment with voltage u^pred; and 2) same-area error units projecting locally onto an error-receiving compartment with voltage u^err (see Fig 1 a3, inset). Each representation unit additively (Fig 1 a2, left) integrates both streams in its somatic voltage u^som:

(1)

where C_m is the capacitance of the soma, g_l is the leak of the soma, and {g^rep, g^err} are the conductances of the two dendritic compartments. Intuitively, the somatic voltage u^som is pulled towards u^pred with strength g^rep, towards u^err with strength g^err, and towards zero with strength g_l. In other words, the relative size of these conductances determines how strongly the soma is affected by inputs from other prediction and error units, respectively. For , the soma of a representation neuron mostly encodes feedforward inputs from other prediction units, behaving like the units in an ANN. Thus, in the limit , our network dynamics can reproduce an ANN trained with BP; see Section 6.2 for an extended discussion. From these dynamics, one can further see that representation neurons have an effective membrane time constant . For simplicity, we set the leak potential E_l to zero for all neurons in our model; this can easily be included by adding a term g_l E_l to all somatic dynamics.

2.5. Temporal dynamics

In our model, the neuronal output is an instantaneous firing rate r(t). Across nearly all instances in the literature, r(t) is modeled as a function of the current membrane potential u(t) only (e.g., [24,29,30,60]). However, this does not represent the temporal expressiveness of real neurons well, whose responses also depend on the dynamics of stimulating currents [61–63]. This effect can, to some degree, be captured by the ‘prospective (!) coding’ mechanism [43,44,63], where firing rates are functions of u(t) and instantaneous rate of change . This means that the instantaneous firing rate is

(2)

where is defined as the prospective voltage with time constant . Various biological mechanisms can explain how neurons may compute ; e.g., coupling of adaptation currents as in the Izhikevich or AdEx models (see Section 6.3 in [44]).

The time constant determines the neuron’s temporal horizon for anticipating its own future firing rate: for fast information transfer, neurons may have approximately equal prospective and membrane time constants, , such that the unavoidable lag introduced by the slow membrane is compensated by the prospectivity of the output. On the other hand, choosing smaller can implement neuron-wise short-term memory, enabling the network to solve temporal tasks (cf. DMS task, Fig 2). More generally, these time constants can be learned, enabling networks to adapt their memory to solve complex spatio-temporal tasks [43,44]. For simplicity, we restrict ourselves to fixed in this work. We denote the prospective firing rate of representation and error units as and respectively. Note that we use linear activations for error neurons in our model (see below).

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Fig 2. Error neuron microcircuits learn motor control and memory tasks.

a1) Setup of cart pole task. The objective is to control the inverted pendulum such that and are reached. a2) A microcircuit of size [4–1] successfully learns to stabilize the pendulum provided inputs . We show the number of failed runs for the linear-quadratic regulator (LQR) and two error neuron microcircuits with different activation functions and local variability level (one ‘ideal’ and one ‘realistic’ microcircuit setting). We show median and quartiles over 10 seeds. a3) Example trajectories for the ‘realistic’ microcircuit after training. The cart pole is reset every 2 s. b1) Delayed match-to-sample (DMS) task and network model. We show two of the four conditions with corresponding targets: (solid line) and (dotted). The time series is fed into a two-area network, each with [2–1] representation and error units. For both conditions, respective targets are marked by a red cross. b2) Accuracy during validation and testing for our model and an equivalent RNN trained with BPTT; median and quartiles over 10 seeds. b3) Example dynamics for and after training, showing the representation neuron rates and . Note the different length of the delay for each of the two conditions. The network makes use of the slow membrane as short-term memory.

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2.6. Error units: morphology

Error units encode a neuron-specific error signal in their soma, which they provide to local representation units. In order to encode useful errors, they need to integrate backpropagated errors from other areas with information from local representation units, in the form of the derivative of the activation . The integration of both signals is realized by one somatic and two dendritic compartments in each error neuron: one receiving errors from other areas, and a representation-receiving one for the reconstruction of . Assuming that error units are L2/3 PYR, their morphology may also follow the examples in Fig 1 a1.

However, the error backpropagation algorithm requires that the soma integrates the product of the two dendritic signals – as opposed to their sum, as in L5 representation units. Error units could use gain modulation to implement such multiplicative dynamics, similar to the mechanism observed in L5 PYR [58,59,64]. Note that it is also possible that error neurons only have a single dendritic compartment, which would be gain modulated by , without explicit encoding of the rate derivative in a dendritic compartment.

2.7. Error units: neuron dynamics

The soma of an error unit e^som multiplicatively integrates its dendritic compartments (Fig 1 a2, right),

(3)

where g^den denotes the effective conductance coupling of both compartments to the soma, e^rep is the voltage of the compartment receiving local representations, while e^err is the compartment receiving error signals from other areas (cf. Fig 1 a3, inset). Intuitively, the somatic voltage e^som is pulled towards with strength g^den, and towards zero with strength g_l. The modulation factor can be reconstructed in the dendrite using temporal noise propagating in the network, and the use of non-linear dendrites (see Methods, Section 6.4). For simplicity, we summarize the effective computation performed in both compartments as .

Note that in our model, error signals are explicitly projected across areas, implementing the BP step (see Equation 6). This differentiates our model from other PCNs [21–25,29], where errors are computed within each area as a local difference between top-down prediction and bottom-up stimulus. As we will demonstrate, this has benefits in terms of performance (Section 3.2) and biological plausibility (Section 4.2).

In this work, we do not address the handling of signed error signals. As neuronal outputs can only be non-negative, we need to explain how an error unit can communicate both positive and negative errors. This may be addressed by distinct populations for positive and negative errors [65,66], or transformed learning signals [66–70], see Section 4.4. For simplicity, we here assume linear activation functions for all error units, i.e., .

2.8. Synaptic plasticity

We move on to the theory and learning rules for the weights, that is: W projecting representations, B projecting errors, and local weights L^RE and L^ER.

Representation-projecting weights W are the functionally relevant connections in the network, i.e., these weights are trained to minimize the cost of a given task. To model the cortical connectome, we scale connectivity between different pairs of areas in W using the projection density of the Macaque visual cortex, see Fig 1 b1. For learning of W, we use a form of the delta rule:

(4)

where is a conductance-weighting factor, reflecting the coupling between soma and dendrite (see Methods). This is a local error-correcting learning rule based on the product of a postsynaptic error () and the presynaptic rate (). Similarly to the work by [71], this can be interpreted as learning with the dendritic prediction of somatic voltage. Note that the postsynaptic term encodes exactly the error signal required by the backpropagation algorithm at the synapse. In particular, for quasi-instantaneous neurons () in a hierarchical setup with ideal local weights , we have

(5)

recovering the exact structure of error backpropagation (see Methods, equations 6 and 7, and equation 17 for more details on the derivation). For different time constants (), error neurons are able to compensate the lag of their partnering representation units, as described in [44].

Local weights are static in our model, and we assume that representation and error units are coupled by preferential targeting – i.e., there is only a loose one-to-one matching between both populations in each area. To model this, we initialize L^RE and L^ER as identity matrices with uniform noise, and equally for L^ER. In Section 3, we demonstrate how the network benefits from such loose matching.

Error-projecting weights: Exact backpropagation, like many realizations of predictive coding [21–23,25], requires symmetric weights for the transport of representations and errors across areas, i.e., . This is known as the weight transport problem, where two distant areas are required to share their synaptic weight information. Many solutions have been proposed in the literature, such as the Kolen-Pollack algorithm [72], Feedback Alignment [73] or Phaseless Alignment Learning (PAL) [74]. The latter harnesses intrinsic neuronal noise to train error-transporting weights to approximate , similarly to how we reconstruct in the error units. Furthermore, PAL is fully compatible with cortical microcircuit models, as demonstrated in [74]. Due to these two reasons, PAL is particularly suited to solve the weight transport problem in our model. It can be implemented by endowing the error-receiving synapses with the learning rule , where is the noise on the potential of the post-synaptic error unit, is the high-pass filtered, pre-synaptic rate of bottom-up error units, and a scalar factor controlling homeostatic regularization. Note however that for computational feasibility, we here approximate PAL by setting error-projecting weights to a noisy transpose of representation weights (see Methods).

Finally, we stress that all information is processed simultaneously in our network (no phases), all information is locally available at the synapse, and learning is always-on.

2.9. Differences to classical PCN

Due to the equivalence of predictive coding and backpropagation under the ‘fixed prediction’ assumption (see Section 4.3), our model is also an implementation of predictive coding. In particular, for ideal local weights, it is computationally equivalent to classical PC models [29,69], but differs in its inter-area connectivity and neuronal dynamics. Most apparent is that we have two separate prediction and error streams, while classical PCNs have bidirectional projections between error and representation units (see also Fig 7). Why do we implement such connectivity? Our model has a number of advantages:

1. (i) Faster and less noisy: predictions don’t need to go through error units in our model, which would double the number of neurons involved in an inference step – this affects processing speed and adds noise sources.
2. (ii) Simple switching between inference and learning: as the error and prediction streams are only loosely coupled, error signals can be introduced dynamically, without the need to wait for the dynamics of all neurons to re-settle. This may be particularly important for tasks where learning signals are only provided sparsely, such as reinforcement learning.
3. (iii) Signal transport and local weights: in most rate-based PCNs, error units compute the local error as a difference of voltages, requiring (bio-implausible) access to the somatic voltage of representation neurons. Alternatively, errors can be defined on the level of firing rates, requiring then that error units have access to the exact activiation function of representation units. In either case, following the exact derivation of PC dynamics based on gradient descent of an optimization function, it appears that error units serve as linear, one-to-one pseudo-compartments of representation units instead of actual neurons.

In our model, errors are explicitly projected from area to area by the error neurons, without the need to compute differences in every error neuron. These errors are then projected locally to representation units, which make them available at the synaptic site W.

3.1. Motor control and spatio-temporal credit assignment

To demonstrate the capabilities of the error neuron microcircuit model for functional learning, we first train our implementation on two small benchmarks. The first is the cart pole task (Fig 2 a), a classical motor control problem also known as the inverted pendulum. The network receives four inputs: the current position and velocity of the cart (), and angle and angular velocity of the pole (). The aim is to produce a motor output (force F) which achieves an upright position of the pole () and navigates the cart to the position x = 0, as illustrated in Fig 2 a1. The cart pole is initialized with and .

We train the networks to reproduce the motor control of a linear-quadratic regulator (LQR), an algorithm that optimizes control for linear systems, and test performance using a simulation of the non-linear dynamics of the cart pole. Performance is measured by the penalty log, i.e., how often the pole falls or fails to reach the position x = 0. As Fig 2 a2 shows, error neuron microcircuits with five representation and error units each learn to solve the task, on par with the LQR in terms of the penalty log. We demonstrate this both with the ‘ideal’ microcircuit, which has linear neuronal activation and perfect one-to-one matching between error and representation units, as well as the more realistic model, with non-linear activations of representation units, and variability on local weights. See Fig 2 a3 for example trajectories of a trained error neuron circuit.

Next, we show that error neuron microcircuits can solve tasks requiring memory (Fig 2b). We use the delayed match-to-sample (DMS) task [75], equivalent to temporal XNOR. The task is to compare two sequential stimuli to identify if the stimuli match sign ( or ) or have opposing signs ( or ), see Fig 2 b1. The input is presented to the network as a one-dimensional time series, and therefore requires short-term memory to be solved. Task complexity is increased by an intermittent delay period with variable length.

In our model, short-term memory is encoded by the leaky integration of the membrane of representation units, and tunable memory length is achieved by the prospective coding mechanism (see Section 2). Task-specific neuronal time constants can be learned, as demonstrated in [43,44]; however, we here only assume a loose agreement between the timescale of the task and neuronal time constants, and thus set  ms; such a large time constant can be provided by neuronal adaptation [44,63]. Fig 2 b2 shows that our model can solve the task using only six neurons (three representation, three error units); example traces are shown in Fig 2 b3. For comparison, we also train an equivalent layered RNN (two hidden units and one output neuron with self-recurrent connections, in lieu of the leaky integration of our representation neurons). We observe that our circuit performs on par with the RNN trained with backpropagation-through-time (BPTT).

3.2. Explicit error neurons scale to many areas, where dendritic error construction struggles

Our model differs from the bio-plausible models of error learning by Sacramento et al. [30] and dendritic hierarchical predictive coding (dendritic hPC, Mikulasch et al. [25]) through the existence of explicit error neurons, as in classical predictive coding models [29] (albeit without local difference calculation at every area). Here, we compare these models to determine advantages of explicit error neurons.

In Fig 3, we show performance of our model on the Yin-Yang task [76], a non-linear classification task with three classes (visualized in Fig 3a). We compare performance to an ANN trained with BP, and the dendritic microcircuit model of Sacramento et al. [30], in its vanilla form (Sacramento et al. (FA)) and with approximate weight transport (Sacramento et al. (PAL)), the latter of which is analogous to the implementation in our model. As Fig 3 b show, our model performs on par with the dendritic microcircuit model with PAL, and nearly reaches the accuracy of the ANN.

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Fig 3. Error neuron microcircuits learn to solve a non-linear classification task.

a) Visualization of the Yin-Yang dataset, with each x-y position corresponding to one of three classes (yin, yang, dot). b) Validation and final test error of the error microcircuit model with network size [30–3]. We compare to the model by Sacramento et al., implemented with feedback alignment (FA) and approximate weight transport like in our model (PAL); these results are reproduced from [74]. We also show the test error in an ANN trained with BP with equal network size. The shading indicates mean and standard deviation over 10 seeds.

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In Fig 4, we demonstrate that our model scales to training of many areas, and performs on par with an ANN of equivalent size. The networks are trained to learn an input-output mapping recorded from a randomly initialized teacher ANN with the same architecture; therefore, task complexity increases with the number of areas in the networks. This guarantees that the student networks are in principle able to solve the task, but also that good performance requires training of all areas – in particular, it is not sufficient to train only the output weights.

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Fig 4. Error neuron microcircuit trains many areas, on par with an ANN.

We compare learning in our model to an ANN trained with backpropagation, and the dendritic error computation models of Mikulasch et al. [25] and Sacramento et al. [30]. In order to show that our model scales to large networks, we systematically increase the number of areas. The task is to imitate a teacher network of the same architecture, where task complexity increases with number of areas. a) Mean loss on test set (median and quartiles over 10 seeds; performance before training is across models and task complexities). Our error neuron microcircuit performs on par with the ANN, while the dendritic hierarchical PC model of Mikulasch et al. [25] and the microcircuit model of Sacramento et al. [30] perform more poorly when training more complex tasks. b) Alignment between synaptic weight changes in all three models vs. those of the ANN (median over all seeds and areas).

https://doi.org/10.1371/journal.pcbi.1014164.g004

We again compare performance to the model of Sacramento et al. with approximate weight transport, and also include dendritic hPC [25] in these experiments. To ensure a fair comparison, we have generalized dendritic hPC to a non-linear version in this work (see Section S1.2 in S1 Text); however, as we find that the non-linear version performs similarly to the linear model (Fig D in S1 Text), we have used the original model of [25] here.

We find that the two models with dendritic error construction do not scale well to multi-area tasks, indicating that dendritic error construction fails when errors are projected across more than two areas (Fig 4 a). On the other hand, our networks with explicit error representations always perform on par with the baseline, ANNs trained with backpropagation. This is an important result, indicating that current bio-plausible models of dendritic error computation lack the scalability required to explain learning across the cortical hierarchy (see Discussion).

Note that task complexity 5 cannot be fully solved by any network, as even the ANN baseline is only able to reduce the loss by about one order of magnitude. Indeed, training only the last three areas of the ANN achieves about the same performance, with a test loss of . This indicates that the networks are not able to make use of learning in all of their five areas, which explains why all models perform equally on this task (barring the model of Sacramento et al.).

Further, our model differs from the others by incorporating connections between all areas. To ensure that this is not the reason of its higher performance, we have repeated the experiment with strictly hierarchical connectivity only, see Fig A in S1 Text. Indeed, the networks with strictly hierarchical connectivity perform equally well.

In Fig 4 b, we show the alignment between weight dynamics of the different models compared to the layered ANN. In the models of [25] and [30], larger deviations from BP are correlated with decreased performance. Such deviations can be explained, e.g., by 1) violation of the ‘fixed prediction’ assumption (see Section 4.3), and 2) imperfect lateral inhibition, generating ‘wrong’ error signals in the dendrites. At least for the model of Mikulasch et al., we have identified the latter to be the cause of the large alignment angles and drop in performance, cf. Fig B in S1 Text.

Curiously, in our model, we do not observe a correlation between performance and weight trajectories compared to BP. This applies even for the strictly hierarchical implementation (Fig A b in S1 Text), hinting that neural dynamics do not need to obey the ‘fixed prediction’ assumption for successful learning (cf. Section 4.3).

3.3. Networks benefit from loose matching of error and representation units

An essential feature of our model are the populations of error and representation neurons in each area. Here, the precise one-to-one matching of prediction and error neurons usually found in PCNs is replaced by preferential targeting, where local weights are not restricted to an identity matrix. As we will demonstrate, this does not only improve biological plausibility, but also impacts task performance.

We train a two-area network representing the early visual areas (e.g., V1 and V2), with 784 (28 × 28) representation and error neurons each in V1, and a variable number of n_R and n_E neurons in V2; see Fig 5a. Representation neurons are tasked to predict activity in V1 matching the label provided as fixed top-down input. This deterministic, generative network is therefore an implementation of supervised predictive coding. The images fed into the network are 28 × 28 images of the MNIST dataset (one image per digit class), while top-down latent features are provided through one-hot encoding of the corresponding classes.

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Fig 5. Implementation of predictive coding, and stability under variation of neuron populations.

We demonstrate that our model accommodates predictive coding using a generative task, where the top-down input (‘latent feature’) is a digit [0,...,9], and the desired prediction a 28 × 28 image from the MNIST dataset. a) Network setup. As opposed to previous tasks, the network is inverted, thereby implementing stimulus prediction. V1 is modeled with 784 representation and error neurons each, whereas in V2, the number of representation units n_R and error units n_E can be varied. b) Test loss for different n_R, n_E (mean and stdev over 10 seeds). Performance generally increases with larger populations, but is stable even for . Note that the networks benefit from additional representation units (n_R > n_E, lower right). c) Example predictions of the digit ‘8’.

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As Fig 5 b & c show, the networks successfully learn to predict visual stimuli given a latent feature. To make sure that the network is actually required to propagate errors, we disabled learning of W from the top-down latent feature to V2, and indeed find drastically reduced performance with a test loss of 40.3 ± 0.4 (n_R = 1000 = n_E). Finally, an ANN trained on the task achieves similar performance to our model, for which the test loss is 9.2 ± 0.1 (n_R = 1000).

As one expects, increasing the number of representation and error units in our model leads to better predictions (diagonal in Fig 5 b). Interestingly however, the networks’ performance is also stable under variation of population numbers. For example, for all combinations of and , the mean square error (MSE) is approximately the same.

Notably, additional representation units are more useful than extra error units: For a given number of error units, increasing the pool of representation units always improves performance, whereas more error neurons are only beneficial if there already is a sufficient number of representation units (n_R ≥ n_E).

Why does relaxing of strict one-to-one matching not deteriorate performance, but in fact improve it for n_R > n_E? Of the n_R representation units, only n_E effectively receive an error signal, and thus learn; but while the remaining (n_E−n_R) representation units do not learn, they still increase the dimensionality of the represented data. Similar to a high dimensional feature space in a support vector machine, such higher dimensional representations can be exploited by downstream areas for improved learning (see S1.7 in S1 Text for details).

These results imply an experimental prediction, where a larger pool of representation units is generally favored over a larger number of error units (see Discussion).

3.4. Error neuron microcircuits are robust to neuron variability and ablations, unless error signals become too noisy

To assess the impact of parameter variability and ablations on our model, we again consider the generative MNIST task of Fig 5. First, we model variability of neuron activations in a network with [784–1000] representation units, and the same number of error neurons (Fig 6 a; note that error units still are linear, ). We introduce neuron-wise variability to the slope and offset (bias): activation slopes are noised by sampling from , where the standard slope used in the other experiments corresponds to 1. Similarly, offsets are noised using .

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Fig 6. Effects of neuron variability and ablations under the generative MNIST task.

a) We model neuron variability by adding parameter variability to the offset and slope of each neuron’s activation function. We sample from a normal distribution with standard deviation and mean 1 and 0 for slopes and offsets, respectively. Adding variability only to representation units (orange) shows their resilience even to large variability (e.g., ). However, as the networks rely on correct errors to be transported, they display high sensitivity to additional variability on error neuron activations (blue). b) We extend the setting of Fig 5 to three cortical areas (V1, V2 and V4), and train two implementations of our model: one using strictly hierarchical connectivity, and one using the visual cortex connectivity with skip forward and backward connections (Fig 1 b1). After initial training, we ablate connectivity from V2 to V1, and disable learning of new weights between these areas. While the hierarchical model is now unable to solve the task, the visual cortex model makes use of the skip connections to recover its performance. All points show mean and standard deviation over 10 seeds.

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Fig 6 a shows that network performance is rather stable under variability of representation neurons, . This can be explained by the algorithmic power of the error backpropagation algorithm: as long as errors are calculated correctly (no variability on ), representation unit weights are trained to solve the task, independent of their activation function. Only for very large variability, the representation units are unable to learn the task in general, e.g., due to vanishing gradients caused by large slope factors (for which for most somatic voltages).

Variability on the error units on the other hand represents a fundamental challenge: In that case, representation units may possess the capabilities of solving the task, but are not guided to a solution due to faulty error signals. We see this reflected in the failure to train already at small levels of variability on . In Fig C in S1 Text, we demonstrate that this can be fully attributed to offset variability – the results can be interpreted using the framework of sign-concordant feedback alignment [77], for which we refer to the Discussion.

Finally, we study how ablations affect our microcircuit model (Fig 6 b). In strictly hierarchical models of cortex, the disruption of synaptic weights between two areas fatally hinders prediction and learning. Due to our inclusion of realistic connectivity based on visual cortex, other areas may learn to take on the role of the ablated connectivity. We extend our model implementing the generative MNIST task to three areas, modeling V1, V2 and V4 with [784-500-500] representation neurons, and an equal number of error units. After initial training using the full connectome, we ablate the weights W projecting representations from V2 to V1, and disallow growth of new synapses between these areas. Upon continuing training, we see that the synapses projecting from V4 to V1 are able to take on the role of the ablated connections, and fully recover predictive performance. As expected, a strictly hierarchical implementation fails to recover, as there is no way for V2 or V4 to project to V1 anymore.

4.1. Experimental evidence and predictions

By integrating our novel mechanisms with aspects from previous works, we are able to address a wide range of bio-plausibility issues of the ‘backpropagation and the brain’ hypothesis; for a comprehensive list, see Table 1. In the following, we will describe concrete measurable correlates which will be able to differentiate our hypothesis from other models of learning in cortex.

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Table 1. Translation of error backpropagation to bio-plausible model. We describe issues of bio-plausibility in ANNs and the BP algorithm (first section), and general issues of models of learning in cortex (second section). The last entries show important issues not addressed in this work, and potential solutions.

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Generally, our model allows for two possible configurations: the (generative) predictive coding implementation, where a top-down stream aims to predict stimuli, and a classifier configuration with inverted streams (Fig 1 b2). Cortex may implement either configuration; however, due to behavioral and neuroscience studies supporting top-down prediction in the cortex, we will focus on the generative configuration in this discussion.

4.1.5. Inter-area connectivity.

Our model differs from classical PCNs (e.g., the Rao-Ballard model [29]) by its inter-area connectivity scheme, see Fig 7. Our configuration has multiple benefits in terms of computational power and biological plausibility of the neuronal calculations, as we describe in Section 4.2.

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Fig 7. Connectivity in our model and classical PCNs.

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In detail, our model predicts two separated streams of errors and predictions. Various studies have investigated preferential targeting of dendritic compartments by top-down and bottom-up projections [14–16,42,55] and mapped out distinct feed-forward and feedback pathways in the cortical hierarchy [52].

Concretely, if we assume that representation units are PYR in L5, and errors are encoded in L2/3, this predicts a top-down stream of L5 → L5 projections, and an opposing error stream projecting L2/3 → L2/3. This prediction concerns the effective functional connectivity between areas, regardless of the underlying anatomical implementation (see S1.5 in S1 Text for an extended discussion on a recent experimental result).

Alternatively, our model can be modified such that representation units do not project locally to error units in the same area, but to error units in upstream areas; see S1.6 in S1 Text.

4.2. Embedding into cortical models of learning and predictive processing

Most implementations of error learning in the cerebral cortex can be grouped into two categories: those with error neurons (as in the classical Rao-Ballard model [29]), and models where errors are constructed locally on the dendrites of representation units [25,30]. As of now, dendritic error construction is not well tested as a model of the cortical hierarchy, and scaling to multi-area networks remains to be proven. In [30], Sacramento et al. successfully trained a three-area network on the MNIST digit classification task; however, the same performance can be reached by leaving out the first area [97], making it unclear if useful error signals have reached the early area in these results.

In this work, we show that explicit error neurons facilitate more stable learning across cortical areas, as dendritic error construction is susceptible to parameter variability, caused by imperfect lateral inhibition. In short: explicit error neurons allow errors to be propagated through the network and then sent to representation units, while dendritic error construction (like classical predictive coding) requires the calculation of a difference of activity in every area. It is precisely this difference which is propagated as the error; under imperfect weights, this reconstruction of useful error signal fails, as Fig 4 a shows. Another important advantage of explicit error neurons is their increased computational power, as reflected by the ability to solve the spatio-temporal credit assignment problem by approximating backpropagation-through-time (BPTT); see Section 4.3 and [44].

4.3. How closely do we need to implement ‘backpropagation and the brain’?

BP is the algorithm underlying the deep learning revolution, and in practice currently the only way ANNs are trained to solve complex tasks. This is because BP propagates local errors across all areas, tailored to each neuron according to its contribution to task performance. While it is clear that cortex does not implement the error backpropagation algorithm exactly as it is formulated in machine learning, one may explore the hypothesis that some approximate version can explain learning [1,2]. In our model, backpropagation is demonstrably obtained in the limit , where error and prediction pathways decouple, in conjunction with strictly hierarchical connectivity and identity local weights (Section 6.2). While the inclusion of such deviations from an ‘ideal’ implementation of BP is motivated by biological plausibility, it is important that such properties can also be interpreted as features, e.g., as implementations of advanced machine learning techniques. Thus, we now discuss how both non-hierarchical connectivity and non-weak nudging () can fit together with backpropagation on a firm theoretical understanding.

4.4. Future work

Several issues remain unaddressed in this work, some of which we have highlighted in the previous sections. Notably, we have tested our model on supervised benchmarks, which require external target labels. However, by replacing the loss function, our model can also accommodate other learning schemes, where latent representations are formed purely by observation, or only sparse rewards are provided.

Aditionally, our model retains limitations in terms of biological plausibility. First, our error units have linear activations, i.e., they can produce both excitatory and inhibitory outputs. This issue has been discussed in the literature, with a range of possible solutions based on separate populations for negative and positive errors [65,66], error signals coded with respect to a bias, or transformed errors [66–70]. Second, synapses in our model can change their weights from excitatory to inhibitory, and vice versa, in contradiction with biology (Dale’s law). Recent works have successfully trained networks of purely excitatory and inhibitory populations [108,109], but rely on standard training libraries without a mechanistic explanation of the error propagation. It remains an open question how to model error neuron microcircuits observing Dale’s law. Third, our model remains a rate-based model. An extension to spikes can be conceived, guided by a recent spiking implementation [78] of the dendritic error construction model [30].

Further, our model singles out the interconnectivity of feedback-projecting PYR in L5 and feed-forward-projecting L2/3, excluding other cell types and projections both within the cortex and with other brain structures. While this is motivated by the high degree of connectivity between L5 and L2/3, incorporating other known projections into our model implementing the algorithmic core of backpropagation could lead to insights. Particularly relevant projections absent from our model but abundant in biology are lateral (recurrent) projections between PYR of the same layer. Thalamo-cortical projections from higher-order thalamic nuclei also seem to be particularly pertinent, given their strong influence on L5 and L1 [110] and their computational relevance [111–116]. Finally, our work doesn’t account for the full complexity of neuronal and in particular dendritic temporal dynamics. Future work may investigate fast top-down prediction and slower error signaling, which produce distinct time scales in experimental signatures [108].

6.1. Bio-plausibility of the error backpropagation algorithm

In a hierarchical ANN with layers, representations are projected through weights , and layer-wise errors can be obtained recursively from a target provided to the output layer N. The error backpropagation algorithm, applied to such an ANN learning a non-temporal task, consists of two computational steps: the backpropagation of errors,

(6)

and the weight update rule

(7)

Implementing these computations into a bio-plausible model carries several obstacles [1,2]. Our model solves the most pertinent ones in the following way, see Table 1.

6.2. Neuronal dynamics and learning rule

Our microcircuit model implements Equations 6 and 7 in the following way. We define a network with areas, equivalent to layers in the context of ANNs. In our model, we relax the strict hierarchical connectivity, and allow for connections between all areas.

The soma dynamics of the population of representation units in area ℓ, written as a vector , are

(8)

that is, a leaky integration of the prediction- and error-receiving compartments’ voltages and . Compartments are modeled as

(9)

where represents the matrix of weights connecting area k to ℓ. sums all inputs from representation units in other areas (but not from the same area), whereas is formed from output of error neurons in the same area.

The neuronal output is the ‘prospective’ rate [43]

(10)

with a neuron-wise parameter, called the prospective time constant. For quasi-instantaneous transmission, we set . For tasks requiring memory, we encode short-term memory through slow neuron responses, for which (see Section 6.5 and 6.6).

Error units follow the dynamics

(11)

with

(12)

Note that does not carry the units of a voltage, but rather , i.e., it is unitless. Thus, the summand in Equation 11 is a current, and can be added to the somatic integration.

For learning, we need to inject a target signal into the network. We do so in the form of a target rate r^tgt, which is provided to area N (‘output layer’) following slightly modified dynamics:

(13)

This means that the representation neurons in area N communicate their rate locally to error units, which calculate the difference between r^tgt and . Error neurons also implement prospective rates,

(14)

and we set all error neurons to be instantaneous, with for hidden areas; for the output area. Note that in all of our simulations, we choose a linear activation for error units, .

By plugging Equation 12 into Equation 11, we can see that encodes the error in each area:

(15)

In a hierarchical setup with ideal local weights, , this implements Equation 6, under the assumption that (see below).

The dynamics of the representation-projecting weights are given by

(16)

where by comparison with Equation 4 we can identify the total conductance as . By rewriting , we obtain

(17)

For identity local weights and hierarchical connectivity, we find that this implements Equation 7.

However, there is one additional caveat to the exact correspondence of our model and backpropagation: due to the coupled compartments, the soma of representation units does not only encode external inputs, . Instead, errors are added to somata in every area, confounding the activity of all units as a combination of inputs and errors due to the recurrency in the network. BP is only strictly equivalent to our model and predictive coding implementations under the ‘fixed prediction’ assumption, i.e., where the feed-forward activity needs to be saved to be used during learning.

This is reflected in the angle of alignment between in ANNs and our model (cf. Fig 4 b), which tends to be larger than zero. This can be ‘remedied’ by decreasing the influence of the error-receiving compartment on the soma, , a limit also known as weak nudging [105]. However, this also shrinks the learning signal, as evident from Equation 17, which in turn leads to higher variability sensitivity and is less faithful to tentative in-vivo evidence [83]. Instead, we keep g^err as a free parameter; as our results show, our model performs on par with an ANN even for large nudging parameters, implying that neuron activities do not need to exactly follow those of feed-forward ANNs for successful learning.

6.3. Network connectivity

6.3.2. Weight symmetry.

In our networks, errors are projected through weights . Exact error backpropagation requires symmetric weights between two areas, i.e., . While networks can learn even with non-symmetric, static feedback weights (Feedback Alignment, FA [73]), this has been shown to not scale well to complex problems, requiring more neurons to successfully learn a task [74,96,118–120].

An efficient mechanism to learn symmetric weights between two areas is Phaseless Alignment Learning (PAL) [74]. PAL uses temporal noise as an information carrier, allowing to simultaneously learn all error projection weights , without disruption of representation unit weights . Note that PAL makes use of the same principles as the mechanism for local reconstruction of proposed in this work, which we explain in Section 6.4.

In [74], we have shown that PAL is able to efficiently train the dendritic microcircuit model of Sacramento et al. [30]; however, its simulation on conventional hardware is time-consuming due to the requirement of fine-grained simulations of temporal noise.

For computational feasibility, we therefore approximate PAL by setting error projecting weights to the transpose of representation unit weights, and adding noise to mimic the approximate degree of alignment typically found in networks using PAL. We also need to consider that the numbers of error and representation units are not required to be equal in our network. To account for this, we pad the weight matrix with zeros: for example, is an n_R × n_E matrix (n_R: number of representation units in area ℓ; n_E: number of error units), with the ‘diagonal’ entries set to 1, and all others set to 0. For example, for n_R = 2 and n_E = 3 in area ℓ, we have

After every time step, error projections are set to , with sampled once (i.e., at the initialization of the network).

6.4. Non-linear dendrites enable local computation of in error units

Here, we explain how the error neurons reconstruct in their dendritic compartment. To simplify notation, we use a generic voltage u in this subsection, acting as a stand-in for the prospective voltage of the representation units.

Error neurons are able to estimate using biologically plausible ingredients: temporal noise, which is ubiquitous in the brain, and simple functions (high-pass filtering, gating and temporal averaging) that a dendrite can perform.

To start, we take a local population of representation neurons with membrane potential vector u, and add noise fluctuations for each neuron, , where can be, e.g., white noise; note that any noise with sufficiently short correlation time suffices. In order for the neurons to encode a useful representation, the signal-to-noise ratio needs to sufficiently high, meaning that . Coincidentally, this is a useful property which we exploit in the following:

The error neuron receives the noisy output of the representation unit at its dendrite,

(18)

where we have expanded the rate to first order in small noise . Next, we aim to isolate the component of r governed by noise (second term). Note that u is driven by stimuli and error signals in the network, which are of low frequency compared to the fast noise . Therefore, we hypothesize that the error neuron dendrites integrate a high-pass filter, leading to

(19)

By applying non-linear gating onto its input, e.g., in the form of a rectified linear (relu) function, and averaging over noise samples, the dendrites compute

(20)

This shows that a rescaled version of can be computed locally in the error neuron dendrite, for sufficiently smooth activation functions. See Fig 8 for simulation results for one pair of representation and error units.

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Fig 8. Temporal noise and non-linear dendrites enable the reconstruction of in error neurons.

A representation unit encodes a signal voltage u for T_pres = 100 dt, and white noise is sampled at every dt. After high-pass filtering with , the mean dendritic activity accurately tracks for different activation functions. The overall normalization of each curve has been rescaled for ease of illustration.

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6.5. Simulations

We simulated the error neuron microcircuits using the Euler forward method with discrete time steps dt, implementing the neuronal dynamics of Equations 8, 9, 10, 12, 11, 13, and 14 and the plasticity of Equation 16. Note that neuronal and weight dynamics are applied during all time steps, without phases or information buffering. The implementation builds on the code of [74], simulating the model of dendritic error microcircuits [30], see Methods in [74] for details. In all experiments, all weight matrices are fully connected, and we allow voltages to settle during a brief settling phase of several dt. Data is fed into the networks as pairs of input and targets (supervised training), where each data point is presented for T_pres time steps. As opposed to the model of Sacramento et al., our microcircuit takes target rates r^tgt instead of voltages (Equation 13, see also below).

For parameter values, see Section 6.6.

6.6. Simulations parameters

For all simulations, we set the resting potential to E_l = 0. We set the capacitance as C_m = 1, thus conductances have units ms^–1. All networks (error neuron microcircuit, ANN, Sacramento et al. and dendritic hPC) are trained with vanilla gradient descent with batch size 1. All simulations were performed with 10 different random seeds. For all parameters, see Tables 2 and 3.

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Table 2. Parameters for microcircuit model simulations.

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Table 3. Parameters for microcircuit model simulations (cont’d).

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For the simulation parameters for the Yin-Yang task (Fig 3), see [74]. All values were chosen the same, with special parameters and noise on sampled from for our error microcircuit model.