A receptor for serotonin

Serotonin (5-hydroxytryptamine, 5-HT) acts on at least fourteen receptor types. One of them, the 5-HT_2A receptor, is especially dense on the large pyramidal neurons of cortical layer 5, whose outputs connect cortical areas to each other and to deeper structures. Classic psychedelic substances, including psilocin (the active form of psilocybin), LSD and mescaline, are agonists at this receptor, and in human studies blocking it with an antagonist such as ketanserin largely prevents their subjective effects (Vollenweider et al. 1998; Nichols 2016).

How tightly a molecule binds

The basic quantitative tool of receptor pharmacology is the binding curve. If a ligand binds reversibly to a receptor, then at equilibrium the fraction of receptors occupied depends on the free concentration $c$ and the dissociation constant $K_d$, the concentration at which half the receptors are occupied:

$$\text{occupancy} = \frac{c^n}{K_d^{\,n} + c^n}$$

This is the Hill–Langmuir equation; $n$, the Hill coefficient, is 1 for simple one-to-one binding and greater than 1 when binding is cooperative. A competitive antagonist shifts the curve to the right, raising the apparent $K_d$ by a factor $1 + I/K_i$.

# Receptor occupancy by the Hill–Langmuir equation, in units of the
# dissociation constant Kd (concentration / Kd). No doses: the numbers are
# dimensionless and say nothing about any amount a person might take.

def occupancy(c, kd=1.0, n=1.0):
    """Fraction of receptors bound at free ligand concentration c."""
    return c ** n / (kd ** n + c ** n)

print("c/Kd   occupied (n = 1)   occupied (n = 2)")
for c in (0.1, 0.3, 1, 3, 10):
    print(f"{c:4g}   {occupancy(c):16.2f}   {occupancy(c, n=2):16.2f}")

# A competitive antagonist at concentration I with its own constant Ki
# shifts the agonist curve: apparent Kd' = Kd (1 + I/Ki).
for i_over_ki in (0, 1, 9):
    kd_app = 1 + i_over_ki
    print(f"antagonist I/Ki = {i_over_ki}: apparent Kd × {kd_app:g}, "
          f"occupancy at c/Kd = 1 is {occupancy(1, kd_app):.2f}")
# c/Kd   occupied (n = 1)   occupied (n = 2)
#  0.1               0.09               0.01
#  0.3               0.23               0.08
#    1               0.50               0.50
#    3               0.75               0.90
#   10               0.91               0.99
# antagonist I/Ki = 0: apparent Kd × 1, occupancy at c/Kd = 1 is 0.50
# antagonist I/Ki = 1: apparent Kd × 2, occupancy at c/Kd = 1 is 0.33
# antagonist I/Ki = 9: apparent Kd × 10, occupancy at c/Kd = 1 is 0.09

Occupancy is not effect. What a receptor does when occupied depends on the ligand’s efficacy and on which intracellular pathways it favours, a property called functional selectivity or biased agonism. This is one reason why molecules that bind the same receptor can have very different effects, and why binding numbers alone cannot predict experience (Nichols 2016).

Measuring the richness of a signal

The “entropic brain” hypothesis uses ideas from information theory. Two measures are common. Shannon entropy asks how unpredictable the symbols of a signal are, given how often each occurs. Lempel–Ziv complexity asks how many new patterns appear as one reads through a signal; a perfectly regular signal can be described with few patterns, a random one needs many.

import math, random

def lz76(s):
    """Number of phrases in the Lempel–Ziv (1976) parsing of a 0/1 string."""
    i, c, n = 0, 0, len(s)
    while i < n:
        k = 1
        while i + k <= n and s[i:i + k] in s[:i + k - 1]:
            k += 1
        c += 1
        i += k
    return c

def binarise(x):
    m = sorted(x)[len(x) // 2]                      # split at the median
    return "".join("1" if v > m else "0" for v in x)

def entropy_words(s, L=3):
    """Shannon entropy (bits) of the 3-symbol words in the string."""
    counts = {}
    for i in range(len(s) - L + 1):
        counts[s[i:i + L]] = counts.get(s[i:i + L], 0) + 1
    tot = sum(counts.values())
    return -sum(c / tot * math.log2(c / tot) for c in counts.values())

rnd = random.Random(7)
n, fs = 2000, 250                                  # 8 s at 250 samples per second
t = [i / fs for i in range(n)]
signals = {
    "10 Hz rhythm":            [math.sin(2 * math.pi * 10 * u) for u in t],
    "10 Hz rhythm + noise":    [math.sin(2 * math.pi * 10 * u) + 0.8 * rnd.gauss(0, 1) for u in t],
    "noise only":              [rnd.gauss(0, 1) for _ in t],
}
norm = n / math.log2(n)                            # LZ count of a random string, roughly
print("signal                  LZ (normalised)   entropy of 3-bit words")
for name, x in signals.items():
    s = binarise(x)
    print(f"{name:22s}   {lz76(s) / norm:14.2f}   {entropy_words(s):10.2f} of 3.00 bits")
# signal                  LZ (normalised)   entropy of 3-bit words
# 10 Hz rhythm                       0.07         1.79 of 3.00 bits
# 10 Hz rhythm + noise               0.84         2.75 of 3.00 bits
# noise only                         1.05         3.00 of 3.00 bits

The output makes a point that is easy to miss: the highest complexity belongs to pure noise. More entropy is not, in itself, a better or richer state. A brain that showed maximum entropy would be a brain whose activity carried no structure at all. Whatever value the measures have depends on comparison with the right baseline and on what else changes at the same time.

The imaging findings

From findings like these Carhart-Harris and colleagues proposed the entropic brain hypothesis: that the quality of conscious states can be indexed by the entropy of brain activity, with ordinary waking consciousness held in a relatively constrained, lower-entropy regime by the integrating activity of the default network, and with psychedelic states as higher-entropy states in which that constraint is relaxed (Carhart-Harris et al. 2014). A later version, “relaxed beliefs under psychedelics” (REBUS), recast this in the language of predictive processing: the substances are proposed to weaken the precision of high-level prior beliefs, letting bottom-up information have more influence (Carhart-Harris and Friston 2019).

The clinical evidence and its limits

The limits of this evidence are substantial and are acknowledged by the researchers. Blinding is very hard, because participants usually know whether they have received an active substance, and expectations are high. Trials are small and short, and the people enrolled are carefully selected: guidelines for research exclude people with a personal or family history of psychotic disorders, and people with significant cardiovascular disease, and provide preparation, supervision throughout the session and follow-up (Johnson et al. 2008). Results under those conditions say nothing about use outside them, which carries the legal and medical risks set out at the start of this chapter.