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In neuronal networks, the dimensions refer to the number of neurons, layers, and connections. Here are some ways to monitor and manage the dimensions:

1. Visualization: Visualizing the network can help to understand the dimensions and how they affect the network's performance. Tools like TensorBoard can visualize the graph, activation, and parameter distributions.

2. Regularization: Regularization techniques such as L1, L2, and dropout can help to reduce the number of neurons and connections or prevent overfitting, which can cause the network to become too complex or too specialized for the training data.

3. Hyperparameter tuning: Hyperparameters such as the number of layers, neurons, and connections can be tuned using grid search or randomized search to find the optimal configuration for the network.

4. Compression techniques: Compression techniques such as pruning, quantization, and knowledge distillation can reduce the dimensions of the network while preserving the accuracy.

5. Monitoring: Monitoring the training and testing accuracy, loss, and performance metrics can help to detect if the network is becoming too complex, overfitting, or underfitting.