Passive Cable Properties of Neurons

Introduction to Passive Cable Properties

Passive cable properties represent the non-voltage-dependent 'electrical skeleton' of neurons. This framework is essential for understanding neural, dendritic, and synaptic integration. The primary objective is to comprehend how the physical structure and electrical parameters of a neuron dictate its ability to process and transmit information.

Learning Outcomes

Neuronal Structural Diversity: Understanding the wide variety of neuronal shapes in the brain, each designed for specific functions.
RC Circuit Representation: Gaining knowledge that a neuron can be effectively described using an RC (Resistor-Capacitor) circuit model.
Passive vs. Active Properties: Distinguishing between passive electrical properties (constant) and active electrical properties (voltage or ion-dependent).
Integration of Inputs: Understanding temporal and spatial integration and how these mechanisms allow neurons to process multiple inputs.

Neuronal Morphology and Structural Diversity

Neurons are complex, three-dimensional tree-like structures. Historically and currently, various techniques are used to visualize this complexity:

Golgi Staining: An older technology discovered by accident. It is still utilized because it labels only a sparse subset of neurons, allowing individual structures like the Neocortex and Hippocampus to be seen clearly.
Thy-1 Mouse: A modern genetic method where a subset of neurons is genetically tagged with Green Fluorescent Protein (GFP).
Monosynaptic Trans-synaptic Retrograde Labelling: Using modified rabies viruses to label pre-synaptic cells. Research by Wertz et al. (2015) in the mouse visual cortex showed that while a single layer 2 pyramidal neuron (red) might have $1,000$ to $5,000$ connections, typically 'only' up to approximately $700$ pre-synaptic cells (green) are labeled.

Catalog of Neuronal Tree Structures

Neuronal shapes vary significantly depending on their location and role:

Spinal cord motoneuron
Retinal ganglion cell
Retinal amacrine cell
Cerebellar Purkinje cell (human): Highly complex and branched.
Purkinje cell (electro-sensing fish)
Thalamic relay neuron
Striatal medium spiny projection neuron
Olfactory bulb granule cell
Locust interneuron
Neocortical L5 pyramidal cell: Characterized by distinct apical and basal dendritic arbors.

Spruston (2008) emphasizes that understanding these complex structures—which underlie thoughts—requires simplifying them into building blocks. It is noted that axonal branches are often much more extensive than the dendritic arbors shown in standard reconstructions.

Passive vs. Active Electrical Properties

Passive Properties

Passive properties are electrical characteristics that remain constant and do not change with voltage or intracellular calcium concentration ( $[Ca^{2+}]$ ).

Membrane Capacitance ( $C_m$ ): This serves to integrate or "add up" flowing currents. Specific membrane capacitance (capacitance per unit area) is generally constant across cells.
Cytoplasmic (Axial) Conductance: Defined as $1/ ext{resistance}$ . It couples different parts of the cell, allowing compartments to send signals to one another via axial currents flowing down the long axis of the cytoplasmic 'tube'.
Membrane Leak Conductance: Mediated by $K^{+}$ , cation, or $Cl^{-}$ channels that are independent of voltage, Na+, or internal Ca2+. These channels allow the membrane capacitance to discharge.

Active Properties

Active properties involve membrane conductances (ion channels) that change based on voltage, time, or intracellular concentrations of $[Ca^{2+}]$ or $[Na^{+}]$ :

Specific channels ( $Na^{+}$ , $K^{+}$ , $Ca^{2+}$ , or $Cl^{-}$ ) that open upon depolarization.
Specific channels ( $K^{+}$ , cation, or $Cl^{-}$ ) that open upon hyperpolarization.
Channels that open specifically when intracellular $[Ca^{2+}]$ or $[Na^{+}]$ levels rise.

The Electrical Model of the Neuron

Experimental evidence shows that real neurons have charging curves nearly identical to a resistor and capacitor in parallel. This RC circuit provides a definitive mathematical model for neuronal behavior.

Key Passive Parameters

Dendritic Level: * $C_m$ : Specific membrane capacitance (per unit area). * $R_m$ : Specific membrane resistance (of a unit area). * $R_i$ : Cytoplasmic (axial) resistivity. * Resting Membrane Potential (RMP): The baseline voltage, which may vary across different parts of the cell. * Space Constant ( $\lambda$ ): Measures the spread of voltage along the cable.
Cellular Level: * Input Resistance ( $R_{IN}$ ): The total resistance of the cell as 'seen' from a recording point. * Total Membrane Capacitance: Calculated as $C_m \times \text{surface area}$ . * Membrane Time Constant ( $\tau_m$ ): $\tau_m = R_m \times C_m$

Cable and Compartmental Modeling

To simulate single neurons, researchers use cable and compartmental representations. This involves:

Reconstruction: Measuring the lengths ( $l$ ), diameters ( $d$ ), taper, and branching patterns of the soma, axon, and dendrites.
Discretization: Chopping the tree into segments (compartments). If a segment has a constant diameter, it is treated as a leaky cable.
Mapping: Converting the branching tree into a mathematical model (dendrogram). Connectivity is mapped, though horizontal line lengths in these diagrams do not represent physical distance.
Circuit Equivalent: Each compartment is modeled as an electrical circuit with: * Membrane capacitance in parallel with membrane leak conductance. * Axial resistance connecting it to adjacent compartments. * Ignore extracellular resistance (it is much lower due to the larger volume of extracellular fluid).

Current Flow Equations

Axial current ( $I_{ax}$ ) flows along the cytoplasm when neighboring compartments have different membrane potentials. Based on Ohm's Law: $I_{ax(n-1 \rightarrow n)} = g_{ax} \times (V_{n-1} - V_n)$

For any single compartment, the total current ( $I_{total}$ ) flowing onto the intracellular surface of the membrane capacitance is: $I_{total} = I_{axial} + I_{channels}$ Where $I_{axial}$ is the sum of currents from the left and right neighbors: $I_{axial} = g_{ax} \times (V_{n-1} - V_n) + g_{ax} \times (V_{n+1} - V_n)$

Axial Charge Spread

When charge is injected into a biological cable (e.g., via a synapse opening a channel):

Positive charge is driven in, leaving negative charge behind on the outside.
This local charge charges up the nearby membrane capacitance, increasing the local membrane potential.
Charge spreads axially down the cable. This process is generally much faster than the discharge through leak channels because axial resistance is significantly lower than membrane resistance.
Voltage Waveforms: At the input site (C), the transient is large and fast. At intermediate distances (B), it is smaller. At far distances (A), the transient is small and slow-rising. Eventually, waveforms coincide at late times once the charge distribution has equalized along the length of the cable.

Temporal Integration and the Time Constant

Temporal summation occurs when charge piles up on the membrane capacitance from inputs arriving at different times.

The Exponential Decay of Voltage

The membrane potential ( $V$ ) discharges exponentially back toward the RMP after an input: $V = V_0 \times e^{-\frac{t}{\tau_m}}$

Long Time Constant ( $\tau_m$ ): Results in significant temporal summation (e.g., $\tau_m = 10\,ms$ ).
Short Time Constant: Results in little temporal summation (e.g., $\tau_m = 1\,ms$ ).

Defining Features of Exponential Decay

The slope is proportional to the negative height.
The quantity falls by a fixed fraction over a fixed time.
Tau ( $\tau$ ) is the $\frac{1}{e}$ life (falling to approximately $0.37$ of the original value).
Half-life: The time it takes for a quantity to halve. $\text{Half-life} \approx 0.69 \times \tau_m$ $\tau_m \approx 1.44 \times \text{Half-life}$

Spatial Integration and the Space Constant

Spatial integration involves the summation of charges from different input locations. The steady-state spread of voltage along a cable is measured by the space (length) constant $\lambda$ .

The Space Constant Formula

If DC (steady) current is injected, it leaks out via membrane channels like water from a leaky hose. The voltage decay along a passive cable follows: $V = V_0 \times e^{-\frac{x}{\lambda}}$ $\lambda = \sqrt{\frac{R_m \times d}{4 \times R_i}}$ Where:

$d$ = diameter.
$R_i$ = cytoplasmic resistivity.
$R_m$ = specific membrane resistance.

At a distance of $1\lambda$ , voltage falls to $\frac{1}{e}$ ( $\sim 0.37$ ). At $2\lambda$ , it falls to $\frac{1}{e^2}$ ( $\sim 0.14$ ).

Dendritic Attenuation and Spines

Real data (Nevian et al., 2007; Larkum et al., 2009) show substantial attenuation of Post-Synaptic Potentials (PSPs) as they travel from the dendrite to the soma:

Somatic Attenuation: An Excitatory Post-Synaptic Potential (EPSP) recorded at the soma can be up to $40$ -fold smaller than the original EPSP near the synapse.
Mechanism: Attenuation is largely due to the capacitance of the rest of the dendritic tree. As charge flows down voltage gradients, it is 'diluted' over the total membrane capacitance, which is hundreds of times larger than the capacitance local to the synapse.
Voltage Equation: $V = \frac{Q}{C}$ . As capacitance ( $C$ ) increases (proportional to surface area), the voltage ( $V$ ) for a given charge ( $Q$ ) decreases.
Dendritic Spines: These small protrusions can account for a significant portion of the neuron's total surface area.

Bibliographic References for Further Study

Kandel & Schwartz: Principles of Neural Science, 5th Ed., Ch. 6 (Passive Electrical Properties).
Squire, Berg & Bloom: Fundamental Neuroscience, 4th Ed., Ch. 5 (Electrotonic Properties) and Ch. 10.
Spruston (2008): Nature Reviews Neuroscience 9:206.
Major et al. (1993): Biophysics Journal 65:423 (Solutions for transients in branching cables).
Trevelyan & Jack (2002): Journal of Physiology 539:623 (Passive cable models of L2/3 pyramidal cells).
Nevian et al. (2007): Nature Neuroscience 10:206 (Properties of basal dendrites).