We showed that the mixed selectivity that is commonly observed in PFC responses can be interpreted as a signature of high-dimensional neural representations. One advantage of high dimensionality is that information about all task-relevant aspects and their combinations is represented in a way that is easily accessible to linear classifiers, such as simple neuron models. The information is distributed across multiple neurons in an ‘explicit’ format16 that allows a readout neuron to implement an arbitrary classification of its inputs. Previous studies have already shown that a linear readout is often sufficient to decode particular task aspects or to perform specific tasks (see for example refs 17, 18). Here, by showing that the neural representations are high-dimensional, we demonstrate that any binary choice task involving the 24 experimental conditions that we analysed could be performed by a linear readout.

One of our main results is that the dimensionality of the neural representations collapses in error trials, indicating that nonlinear mixed selectivity might be important for generating correct behavioural responses. It is tempting to speculate about the causes of this dimensionality collapse. Nonlinear mixed selectivity can change in a way that is compatible with our observations when neurons integrate multiple sources of information, which include those that are relevant for the task and those that are not under experimental control. The change in dimensionality may be caused by the excessive variability of sources that are not task-relevant. In other words, to perform correctly, the brain has to mix nonlinearly the task-relevant sources of information in a way that is consistent across trials. This consistency requires to restrict the contribution of the other sources. This is similar to what has been observed in the premotor cortex, where firing rates tended to be less variable on trials in which the reaction time was shorter19. A theoretical argument (Supplementary Section S.18) shows that neurons with a strong nonlinear mixed selectivity are more sensitive than pure selectivity neurons to the task-irrelevant sources of variability. Nonlinear mixed selectivity is most useful but also most fragile. Pure and linear mixed selectivity, which are more robust, make it possible to decode individually all task-relevant aspects even in the error trials, as observed here.

Although high dimensionality is not strictly necessary for generating rich dynamics and performing complex tasks, it is known to greatly simplify the design of local neural circuits9. Indeed, realizing a complex and rich dynamics is for some model circuits equivalent to solving a classification problem in which the network has to generate a particular output for each input. In these models this is typically realized by training a subset of neurons to respond in a specific way to an external input or to the internally generated activity. This is equivalent to classifying the activity of the input neurons for every time step. In many situations this activity is read out by downstream circuits. In others it is fed back to the neural circuit to affect its dynamics and hence the statistics of future inputs. Especially in the latter situations, the number of input–output functions or classifications that must be implemented by each neuron can be significantly larger than the number of functions required to simply produce the observed final behavioural response, because the neurons are required to generate the proper output for every time step. For this reason, it is often necessary to expand the dimensionality of the neuronal representations of the external sensory input and the internal state. In recent models5,6,7,8,9,10, the dimensionality of the neuronal representations is expanded by mixing in a nonlinear way the different sources of information in a population of randomly connected neurons. The resulting neuronal representations are high-dimensional (see for example ref. 20), like those observed in PFC, and consistent with high dimensionality, the neurons show mixed selectivity which is diverse across time (that is, in different epochs of the trials) and space (that is, across different neurons). Random connectivity in small brain regions has been suggested on the basis of anatomical reconstructions21 and recently observed in the connections from the olfactory bulb to the olfactory cortex22 (see also ref. 14 for a general discussion).

We showed that the recorded mixed selectivity can be useful for the activity to be linearly read out. It is legitimate to ask whether these considerations would still be valid if we consider more complex nonlinear readouts. For example, some of the transformations which increase the dimensionality of the neural representations could be implemented at the level of individual neurons by exploiting dendritic nonlinearities. Our results do not exclude the functional importance of such dendritic processes. They do, however, tend to argue against a scenario where all important nonlinear transformations are carried out at the level of single neurons, a situation where dimensionality expansion could happen in a ‘hidden way’, and the observable representations provided by the neuronal firing rates could therefore be low-dimensional.

Finally, the particular form of redundancy inherited from high-dimensional representations allows neural circuits to flexibly and quickly adapt to execute new tasks, just as it allows them to implement arbitrary binary classifications by modifying the weights of a readout neuron (using, for instance, a supervised procedure like the perceptron learning rule23). In Supplementary Section S.9 we show an example of this flexibility by training a simulated neuron to perform a new virtual task based on the recorded activity. High dimensionality might therefore be at the basis of the mechanisms underlying the remarkable adaptability of the neural coding observed in the PFC13 and, as such, be an important element to answer fundamental questions that try to map cognitive to neurophysiological functions.

In conclusion, the measured dimensionality of the neural representations in PFC is high, and errors follow a collapse in dimensionality. This provides us with a motivation to shift the focus of attention from pure selectivity neurons, which are easily interpretable, to the widely observed but rarely analysed mixed selectivity neurons, especially in the complex task designs that are becoming progressively more accessible to investigation.