Uniform growth and neuronal integration

1. The cable equations were analyzed to determine the effects of two patterns of uniform growth on the passive and active integrative properties of neurons. 2. During uniform isoelectrotonic growth, the diameters of all neuronal processes increase as the square of their increase in length, while the specific electrical properties and branch terminal conditions of the neuron remain constant. All analytic inductive proof is given to show that, for any neuron, uniform isoelectrotonic growth increases the input conductance everywhere by the cube of the growth factor, but leaves the active and passive spread of membrane potential within the neuron unchanged. The spread of membrane voltage is unchanged because this pattern of growth enables both the axial and membrane currents everywhere in the cell to increase by the cube of the growth factor. Synaptic inputs would evoke the same responses in the isoelectrotonically larger cell as in the smaller cell if the total postsynaptic conductance of the synapse increased with the dendritic membrane area. 3. During uniform isometric growth, the diameter and lengths of all processes increase by the same factor, while the specific electrical properties and branch terminal conditions remain constant. This pattern of uniform growth increases the input conductance by the square of the growth factor, and also increases the attenuation, delay, and low-pass filtering of the cell's responses. Voltage attenuation increases with isometric growth because the axial current increases in proportion to growth, while the membrane current increases in proportion to the square of the growth factor. Isometric growth reduces the ability of distal synaptic inputs to affect the membrane potential at proximal integrating sites, even after the synaptic conductance has increased to compensate for the increased input conductance. 4. These two patterns of uniform growth help define the consequences of all types of uniform growth for neuronal integration and responsiveness.