In the neocortex, input patterns are compressed hierarchically. Sensory inputs in the lower levels are combined by higher levels to form abstract concepts. However, there are even more feedback connections ranging from higher levels downward to lower ones. Those are assumed to unroll higher concepts back into lower patterns that could have caused them in the first place.

If patterns could flow up and down through the same synapses, it would be easy to understand how higher patterns cause lower patterns similar to those that could have caused them. In a mathematical sense, traversing the connections in the inverse direction would cause the inverse of the upward transformation.

But synapses are mainly directed. So patterns cannot flow downward through the same synapses. How do the feedback connections still learn transformations similar to the inverse of the upward connections?

  • $\begingroup$ what an excellent question $\endgroup$
    – honi
    Feb 3 '16 at 4:01
  • $\begingroup$ I touched on how these structures can be modelled in this question, however I still need to think about how they can be learned and how to explain this learning. $\endgroup$
    – Seanny123
    Mar 8 '16 at 0:45


There's some points to keep in mind when thinking about this:

  • While it is true that connections in cortex are reciprocal, this does not mean that they are symmetric. Cortical connectivity is quite elaborate and the neurons that send bottom-up signals to the next higher area are not necessarily the same neurons that receive the feedback. The forward signals seem to originate mostly from cortical layers 2/3 and 5, whereas the targets of feedback are primarily layers 1 and 6 (see for example Roelfsema & Holtmaat (2018) figure 2 and Shipp (2007)). On the other hand, an axon that terminates in layer 1 may be contacting dendrites of neurons with cell bodies in other layers, so the story is not that simple (see Larkum et al. (2018)).
  • The connections may not be monosynaptic. Many of the communication between cortical areas happens via the thalamus and other subcortical structures (see the same papers as in the last point).

Learning symmetric connections

I will ignore these two points for now and just assume that a set of neurons in a lower-level cortical area (for example, V1) and a set of neurons in a higher-level cortical area (let's say V2) have reciprocal, monosynaptic connections. So how can a neuron $y_j$ in V2 ensure that its feedback to V1 generates roughly the pattern that excited $y_j$ in the first place? One possibility is to use Hebbian learning. Essentially, if synaptic strength $w_{ji}$ in the forward and $w_{ij}$ in the feedback pathway both grow when $x_i$ in V1 and $y_j$ in V2 fire synchronously / with strong rates, the connections will tend to become symmetric over time.

Instar-outstar learning

Pure Hebbian learning ($\Delta w_{ji} \propto x_i \cdot y_j$) tends to be unstable, so one needs to add a decay term to the weights. This way you arrive at instar-outstar learning (see for example Grossberg (2013), section 1.5), which has been used for exactly this purpose: learning appropriate connections between higher-level categories and lower-level inputs.

The forward connection would perform instar learning ($\Delta w_{ji} \propto y_j \cdot (x_i - w_{ji})$). This means that the weights of all synapses going into neuron $y_j$ compete. Over time, $w_{ji}$ will tend to the mean input vector $\langle \mathbf{x} \rangle$.

The feedback connection would perform outstar learning ($\Delta w_{ji} \propto x_i \cdot (y_j - w_{ji})$). This means that the weights of all synapses going out of neuron $y_j$ are in competition. Note that this may be difficult to implement in a biological neural network, since it means that synapses on different neurons $x_i$ have to compete with each other. Over time, $w_{ij}$ will also tend to the mean output vector $\langle \mathbf{x} \rangle$. Since the input of the forward connection is the output of the feedback connection are identical, the weight vectors should become symmetric.

Oja learning

Note however, that this means the weight matrices are transposes of each other, not inverses. Also, you would need to ensure that different neurons in V2 do not all learn the same pattern, for example by inhibitory connections between them.

One way to overcome both of these problems is to use the Oja subspace rule (see Oja (1992, 1997)): $\Delta \mathbf{W}_{ji} \propto (\mathbf{x} - \mathbf{W} \cdot \mathbf{y}) \cdot \mathbf{y}^T $ This rule ensures that the weight matrix is orthogonal, so that the transpose will be an inverse. However, note that this involves some more competition between synapses on different neurons, so it is not biologically plausible. Note also that the weight vectors will now tend to principle components of the input patterns, not to mean values.

Beyond the simple scenario

These learning rules make it possible to learn symmetric connections in a very simple scenario, assuming directly connected, rate-based neurons. The proofs to show that instar-outstart approximates the mean and Oja's rule performs PCA also assume linear neurons, which is a further simplification. So coming back to the caveats above, in cortex the story is probably not so simple. Still, these principles (Hebbian learning with competition in both directions) may provide the basis for an explanation.

For example, despite caveat 1 Hebbian learning may still be useful since neurons across all layers of a cortical column tend to respond to the same stimuli. So as long as the rates are similar enough, the connections could still learn to be somewhat symmetrical or at least to have the kind of concept unrolling function you have in mind.

Also, while I only talked about rate-based networks, something similar may work for spiking networks as well (see Clopath et al (2010) for a spike-time dependent plasticity rule that can lead to the formation of reciprocal connections).


Connections in cortex may not be directly symmetrical. Making several simplifying assumptions, it is possible to learn symmetrical connections with variants of Hebbian learning.


  • Clopath et al. (2010). Nature Neuroscience 13(3), 344-352. link
  • Grossberg (2013). Neural Networks 37, 1-47. link
  • Larkum et al. (2018). Frontiers in Neuroanatomy 12, 56. link
  • Oja (1992). Neural Networks 5(6), 927-935. link
  • Oja (1997). Neurocomputing 17, 25-45. link
  • Roelfsema & Holtmaat (2018). Nature Reviews Neuroscience 19(3), 166-180. link
  • Shipp (2007). Current Biology 17(12), R443-R449. link

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