Firing together, wiring together

In 1949, in The Organization of Behavior, Hebb proposed that when an axon of cell A is near enough to excite cell B and repeatedly takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency in firing B is increased (Hebb 1949). This is often summed up as “neurons that fire together wire together”. More than two decades later Bliss and Lømo found in anaesthetised rabbits that brief trains of high-frequency stimulation of the perforant path could strengthen synaptic responses in the dentate gyrus for between half an hour and ten hours: long-term potentiation (LTP) (Bliss and Lømo 1973). For the half-century since, the plasticity of synaptic strength has been the central cellular model of learning and memory.

But the synapse is only half of signalling. The signal must also travel along an axon, sometimes a few millimetres, sometimes more than ten centimetres. How fast it travels depends on how thick the axon is and on whether, and how thickly, it is wrapped in myelin.

Myelin: from “fixed insulation” to a plastic structure

In the central nervous system myelin is made by oligodendrocytes: their thin membranes wrap round and round axons, forming segments (“internodes”) separated by bare nodes of Ranvier. The action potential jumps from node to node, much faster and more economically than continuous conduction along an unmyelinated axon. For a long time myelin was thought to be essentially fixed once development was complete. Research over the past fifteen years or so has changed that view: neural activity itself can influence the production of oligodendrocytes and the formation of myelin, a process called activity-dependent myelination (Fields 2015).

Cable theory: how thick should myelin be?

The usual measure of myelin thickness is the g-ratio: the diameter $d$ of the axon divided by the diameter $D$ of the whole fibre (axon plus myelin). The smaller the g-ratio, the thicker the myelin. Analysing myelinated fibres with cable theory in 1951, Rushton concluded that conduction velocity is proportional to fibre diameter, and that for a fixed overall fibre diameter velocity is proportional to

$$v \propto D\, g \sqrt{-\ln g},$$

which is greatest at $g = e^{-1/2} \approx 0.6$ (Rushton 1951). Intuitively: if the myelin is too thin, insulation is poor; if it is too thick, then for a fixed overall diameter the axon itself is too thin. An unmyelinated axon, by contrast, conducts at a speed roughly proportional to the square root of its diameter.

Rushton’s 0.6 matches measurements in some peripheral nerve fibres, but central nervous system fibres usually have higher g-ratios (relatively thinner myelin). In a theoretical analysis in 2009, Chomiak and Hu took spatial (volume) constraints and other factors into account and obtained a theoretical optimal g-ratio of about 0.77 for the central nervous system; weakening the effect of the volume constraint by about 40 % allowed the same approach to recover the peripheral value of about 0.6 (Chomiak and Hu 2009). Note that 0.77 is a theoretical optimum, not a measurement.

A model far simpler than those studies: keep only Rushton’s velocity formula and the $\sqrt{d}$ law for unmyelinated fibres, and compare three cases: fixed overall fibre diameter; a fixed axon with myelin added around it; and a fixed axon with velocity counted per unit of cross-sectional area. All values are relative. The model ignores node and internode geometry, temperature and ion-channel densities, and does not compute the true metabolic cost of making and maintaining myelin; “myelin area relative to axon area” is only a rough index of the cost in space.

# Simplified cable model (relative values). Myelinated (Rushton 1951): v ~ d sqrt(-ln g),
# g = d / D; unmyelinated: v ~ sqrt(d). No node geometry, no true metabolic cost.
import math

def v_fixed_D(g):            # outer diameter fixed (D = 1): Rushton's case
    return g * math.sqrt(-math.log(g))

def v_fixed_d(g):            # axon fixed (d = 1): myelin added around a given axon
    return math.sqrt(-math.log(g))

def v_per_area(g):           # axon fixed, velocity per unit fibre cross-section (D^2 = 1/g^2)
    return v_fixed_d(g) * g * g

grid = [i / 10000 for i in range(3000, 9900)]
for name, f in (("fixed fibre diameter (Rushton)", v_fixed_D),
                ("fixed axon, per unit of cross-section", v_per_area)):
    gb = max(grid, key=f)
    print(f"optimum g for {name:37s}: {gb:.3f}")
print(f"   closed forms: exp(-1/2) = {math.exp(-0.5):.3f}, exp(-1/4) = {math.exp(-0.25):.3f}")

print("\n  g    | v, fixed D | v, fixed axon | v per area | myelin area / axon area")
for g in (0.5, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9):
    a, b = v_fixed_D(g) / v_fixed_D(0.607), v_fixed_d(g) / v_fixed_d(0.8)
    c = v_per_area(g) / v_per_area(0.779)
    print(f"  {g:.2f} |   {a:6.1%}   |    {b:6.1%}     |   {c:6.1%}   |   {1 / g ** 2 - 1:5.2f}")
print("  (columns normalised to their optimum, or to g = 0.80 for the fixed-axon column)")

print("\nscaling with diameter: doubling the axon")
print(f"   myelinated (g fixed): velocity x{2:.2f}; unmyelinated: x{math.sqrt(2):.2f}")
print(f"   10x faster unmyelinated needs 100x the diameter, {100 ** 2:,}x the cross-section")

L, v0 = 0.05, 10.0           # illustrative: 5 cm path at 10 m/s with g = 0.80
for g in (0.80, 0.78, 0.75):
    v = v0 * v_fixed_d(g) / v_fixed_d(0.80)
    print(f"   5 cm path, same axon, g = {g:.2f}: {v:5.2f} m/s, delay {1000 * L / v:4.2f} ms")
# optimum g for fixed fibre diameter (Rushton)       : 0.607
# optimum g for fixed axon, per unit of cross-section: 0.779
#    closed forms: exp(-1/2) = 0.607, exp(-1/4) = 0.779
#
#   g    | v, fixed D | v, fixed axon | v per area | myelin area / axon area
#   0.50 |    97.1%   |    176.2%     |    68.6%   |    3.00
#   0.60 |   100.0%   |    151.3%     |    84.8%   |    1.78
#   0.65 |    99.5%   |    138.9%     |    91.4%   |    1.37
#   0.70 |    97.5%   |    126.4%     |    96.5%   |    1.04
#   0.75 |    93.8%   |    113.5%     |    99.5%   |    0.78
#   0.80 |    88.1%   |    100.0%     |    99.7%   |    0.56
#   0.85 |    79.9%   |     85.3%     |    96.0%   |    0.38
#   0.90 |    68.1%   |     68.7%     |    86.7%   |    0.23
#   (columns normalised to their optimum, or to g = 0.80 for the fixed-axon column)
#
# scaling with diameter: doubling the axon
#    myelinated (g fixed): velocity x2.00; unmyelinated: x1.41
#    10x faster unmyelinated needs 100x the diameter, 10,000x the cross-section
#    5 cm path, same axon, g = 0.80: 10.00 m/s, delay 5.00 ms
#    5 cm path, same axon, g = 0.78: 10.55 m/s, delay 4.74 ms
#    5 cm path, same axon, g = 0.75: 11.35 m/s, delay 4.40 ms

The program shows several things. With fibre diameter fixed, the optimal g-ratio is 0.607, as Rushton found; velocity changes very little between 0.6 and 0.7. With the axon fixed and myelin simply added, thicker myelin is always faster and there is no optimum; but counting velocity per unit of cross-sectional area (white-matter space is limited) moves the optimum to about 0.78. That this simple model lands close to Chomiak and Hu’s 0.77 is not surprising, since both penalise the space a fibre occupies; ours is far cruder than theirs and cannot be used to “confirm” their result. What it shows is a structural point: the optimal thickness of myelin depends on the constraints, and “the thicker the better” does not hold.

The last part of the program takes another angle: on the same axon, lowering the g-ratio from 0.80 to 0.75 (slightly thicker myelin) raises velocity in this model by about 13.5 %, shortening the delay over a 5 cm path from 5.00 to 4.40 milliseconds. Half a millisecond sounds small, but in circuits where signals from several paths must arrive together, differences of this size may matter.

Myelin and timing

Fields and others have suggested that one possible function of activity-dependent myelination is to adjust conduction delays so that signals from different paths arrive in synchrony (Fields 2015). The program above has already shown that small changes in myelin can shift delays by fractions of a millisecond. If adjustments of myelin depend on the timing relations of neural activity, then repeated practice with a consistent timing structure might drive such adjustment more than disorganised activity. The idea is theoretically attractive, but the real mechanisms are far more complex and far from established.

Chapter summary

Hebb’s postulate and long-term potentiation form the classical picture of synaptic plasticity. Research over the past fifteen years or so shows that myelin, too, can change with neural activity: optogenetic stimulation in mice promotes myelination, and blocking the production of new oligodendrocytes impairs motor learning; juggling training in humans changes white-matter DTI measures, but DTI measures are not myelin. Cable theory gives a velocity proportional to diameter for myelinated fibres and an optimal g-ratio of about 0.6 at fixed fibre diameter; including volume constraints, Chomiak and Hu obtain a theoretical value of about 0.77 for the central nervous system, and our simplified model shows that the optimal thickness depends on the constraints. That myelin tunes conduction times and promotes synchrony is an attractive but unproven idea. The next chapter returns to behaviour: how memory fades over time, and how spacing and retrieval change that process.