Neuronal codes for arithmetic rule processing in the human brain

Recruitment of the premotor cortex during arithmetic operations by the monkey

Arithmetic operations are complex mental processes rooted in the abstract concept of numerosity. Despite the significance, the neural architecture responsible for these operations has remained largely uncharted. In this study, we explored the presence of specific neuronal activity in the dorsal premotor cortex of the monkey dedicated to numerical addition and subtraction. Our findings reveal that many of these neural activities undergo a transformation, shifting their coding from arithmetic to motor representations. These motor representations include information about which hand to use and the number of steps involved in the action. We consistently observed that cells related to the right-hand encoded addition, while those linked to the left-hand encoded subtraction, suggesting that arithmetic operations and motor commands are intertwining with each other. Furthermore, we used a multivariate decoding technique to predict the monkey's behaviour based on the activity of these arithmetic-related cells. The classifier trained to discern arithmetic operations, including addition and subtraction, not only predicted the arithmetic decisions but also the subsequent motor actions of the right and left-hand. These findings imply a cognitive extension of the motor cortex's function, where inherent neural systems are repurposed to facilitate arithmetic operations.

Convergent Circuit Computation for Categorization in the Brains of Primates and Songbirds

Categorization is crucial for behavioral flexibility because it enables animals to group stimuli into meaningful classes that can easily be generalized to new circumstances. A most abstract quantitative category is set size, the number of elements in a set. This review explores how categorical number representations are realized by the operations of excitatory and inhibitory neurons in associative telencephalic microcircuits in primates and songbirds. Despite the independent evolution of the primate prefrontal cortex and the avian nidopallium caudolaterale, the neuronal computations of these associative pallial circuits show surprising correspondence. Comparing cellular functions in distantly related taxa can inform about the evolutionary principles of circuit computations for cognition in distinctly but convergently realized brain structures.

Distinct neuronal representation of small and large numbers in the human medial temporal lobe

Whether small numerical quantities are represented by a special subitizing system that is distinct from a large-number estimation system has been debated for over a century. Here we show that two separate neural mechanisms underlie the representation of small and large numbers. We performed single neuron recordings in the medial temporal lobe of neurosurgical patients judging numbers. We found a boundary in neuronal coding around number 4 that correlates with the behavioural transition from subitizing to estimation. In the subitizing range, neurons showed superior tuning selectivity accompanied by suppression effects suggestive of surround inhibition as a selectivity-increasing mechanism. In contrast, tuning selectivity decreased with increasing numbers beyond 4, characterizing a ratio-dependent number estimation system. The two systems with the coding boundary separating them were also indicated using decoding and clustering analyses. The identified small-number subitizing system could be linked to attention and working memory that show comparable capacity limitations.

Neural coding of numerousness

Perception of numerousness, i.e. number of items in a set, is an important cognitive ability, which is present in several animal taxa. In spite of obvious differences in neuroanatomy, insects, fishes, reptiles, birds, and mammals all possess a "number sense". Furthermore, information regarding numbers can belong to different sensory modalities: animals can estimate a number of visual items, a number of tones, or a number of their own movements. Given both the heterogeneity of stimuli and of the brains processing these stimuli, it is hard to imagine that number cognition can be traced back to the same evolutionary conserved neural pathway. However, neurons that selectively respond to the number of stimuli have been described in higher-order integration brain centres both in primates and in birds, two evolutionary distant groups. Although most probably not of the same evolutionary origin, these number neurons share remarkable similarities in their response properties. Instead of homology, this similarity might result from computational advantages of the underlying coding mechanism. This means that one might expect numerousness information to undergo similar steps of neural processing even in evolutionary distant neural pathways. Following this logic, in this review we summarize our current knowledge of how numerousness is processed in the brain from sensory input to coding of abstract information in the higher-order integration centres. We also propose a list of key open questions that might promote future research on number cognition.

Artificial neural network modelling of the neural population code underlying mathematical operations

Mathematical operations have long been regarded as a sparse, symbolic process in neuroimaging studies. In contrast, advances in artificial neural networks (ANN) have enabled extracting distributed representations of mathematical operations. Recent neuroimaging studies have compared distributed representations of the visual, auditory and language domains in ANNs and biological neural networks (BNNs). However, such a relationship has not yet been examined in mathematics. Here we hypothesise that ANN-based distributed representations can explain brain activity patterns of symbolic mathematical operations. We used the fMRI data of a series of mathematical problems with nine different combinations of operators to construct voxel-wise encoding/decoding models using both sparse operator and latent ANN features. Representational similarity analysis demonstrated shared representations between ANN and BNN, an effect particularly evident in the intraparietal sulcus. Feature-brain similarity (FBS) analysis served to reconstruct a sparse representation of mathematical operations based on distributed ANN features in each cortical voxel. Such reconstruction was more efficient when using features from deeper ANN layers. Moreover, latent ANN features allowed the decoding of novel operators not used during model training from brain activity. The current study provides novel insights into the neural code underlying mathematical thought.

Mathematical models of learning and what can be learned from them

Learning is a multi-faceted phenomenon of critical importance and hence attracted a great deal of research, both experimental and theoretical. In this review, we will consider some of the paradigmatic examples of learning and discuss the common themes in theoretical learning research, such as levels of modeling and their corresponding relation to experimental observations and mathematical ideas common to different types of learning.

Quantitative modelling demonstrates format-invariant representations of mathematical problems in the brain

Mathematical problems can be described in either symbolic form or natural language. Previous studies have reported that activation overlaps exist for these two types of mathematical problems, but it is unclear whether they are based on similar brain representations. Furthermore, quantitative modelling of mathematical problem solving has yet to be attempted. In the present study, subjects underwent 3 h of functional magnetic resonance experiments involving math word and math expression problems, and a read word condition without any calculations was used as a control. To evaluate the brain representations of mathematical problems quantitatively, we constructed voxel-wise encoding models. Both intra- and cross-format encoding modelling significantly predicted brain activity predominantly in the left intraparietal sulcus (IPS), even after subtraction of the control condition. Representational similarity analysis and principal component analysis revealed that mathematical problems with different formats had similar cortical organization in the IPS. These findings support the idea that mathematical problems are represented in the brain in a format-invariant manner.

Number selective sensorimotor neurons in the crow translate perceived numerosity into number of actions

Translating a perceived number into a matching number of self-generated actions is a hallmark of numerical reasoning in humans and animals alike. To explore this sensorimotor transformation, we trained crows to judge numerical values in displays and to flexibly plan and perform a matching number of pecks. We report number selective sensorimotor neurons in the crow telencephalon that signaled the impending number of self-generated actions. Neuronal population activity during the sensorimotor transformation period predicted whether the crows mistakenly planned fewer or more pecks than instructed. During sensorimotor transformation, both a static neuronal code characterized by persistently number-selective neurons and a dynamic code originating from neurons carrying rapidly changing numerical information emerged. The findings indicate there are distinct functions of abstract neuronal codes supporting the sensorimotor number system.

Persistent neuronal firing in the medial temporal lobe supports performance and workload of visual working memory in humans

The involvement of the medial temporal lobe (MTL) in working memory is controversially discussed. Recent findings suggest that persistent neural firing in the hippocampus during maintenance in verbal working memory is associated with workload. Here, we recorded single neuron firing in 13 epilepsy patients (7 male) while they performed a visual working memory task. The number of coloured squares in the stimulus set determined the workload of the trial. Performance was almost perfect for low workload (1 and 2 squares) and dropped at high workload (4 and 6 squares), suggesting that high workload exceeded working memory capacity. We identified maintenance neurons in MTL neurons that showed persistent firing during the maintenance period. More maintenance neurons were found in the hippocampus for trials with correct compared to incorrect performance. Maintenance neurons increased and decreased firing in the hippocampus and increased firing in the entorhinal cortex for high compared to low workload. Population firing predicted workload particularly during the maintenance period. Prediction accuracy of workload based on single-trial activity during maintenance was strongest for neurons in the entorhinal cortex and hippocampus. The data suggest that persistent neural firing in the MTL reflects a domain-general process of maintenance supporting performance and workload of multiple items in working memory below and beyond working memory capacity. Persistent neural firing during maintenance in the entorhinal cortex may be associated with its preference to process visual-spatial arrays.