Location of membrane conductance changes by analysis of the input impedance of neurons. I. Theory

If the membrane conductance of a neuron changes, its response to injected current changes. If the change in membrane conductance is restricted to a given subregion of the neuron, that region can be located by analysis of the form of the change in the response of the neuron to current injection. The theoretical basis of this method is rigorously developed in this paper. Location of the membrane conductance change is possible because the higher-frequency components of the injected currents are progressively attenuated by the axial resistance and membrane capacitance of the neuron as they pass from the injection site to electrotonically more distant regions. For the lower-frequency components, this attenuation is less pronounced. Therefore, when a conductance change occurs relatively far from the recording/current-passing electrode, only the lower frequency components of the response are altered, because the higher-frequency components of the current do not even reach that site. When such a conductance change occurs relatively near the electrode, both the lower and the higher frequency components of the response are altered. Treating the neuron as a passive network, the input impedance at a given frequency is simply the voltage response of the neuron at that frequency divided by the current injected at that frequency. This is a complex value, having both magnitude and phase components. The change in the magnitude of the input impedance due to a conductance change occurring distally drops off more rapidly with increasing frequency than that due to a proximal conductance change. In addition, for distal conductance increases the magnitude of the input impedance can increase in the higher range of frequencies. This paradoxical effect is treated in APPENDIX 1. For many neurons an estimate of the electronic location of a conductance change can be made knowing only the change in input resistance, the change in the magnitude of the input impedance at the characteristic frequency (ω₀ = 1/τ₀), and a reasonable estimate of the total electrotonic length of the neuron (L). The sensitivity of the method depends on the electrotonic length of the neuron. The method is most useful in neurons with dendritic trees longer than ~ 0.5 length constants. The dendritic-to-somatic conductance ratio of the neuron does not appreciably affect the forms of the responses. The time constant merely shifts the frequency range of interest. If the changes in conductance produce large changes in input resistance (> 10%) the plots of the change in input impedance as a function of frequency are shifted to the right somewhat from those due to smaller changes, and the estimate of electrotonic location of a conductance change must be corrected for the size of the change in input resistance. This method is valuable, not only theoretically, but also experimentally, because in some cases it may be the only way to determine the electrotonic location of active synapses on a single neuron. The injected currents can be in the form of sine waves, current steps, or impulses. Each form of test current has its own set of advantages and disadvantages, which are discussed. The major drawback of this method is the requirement for relatively long-lasting changes in conductance. Location of conductance changes shorter than a time constant or two in duration would not be possible. The practical application of intracellular sine-wave impedance measurements is developed in the following paper.