A brain in the evidence cone

A fish moves through murky water. Its senses receive an ambiguous clue. It moves again, changes what reaches its receptors, combines the new clue with a memory, and turns toward a hiding place. Later, as the animal rests, activity revisits parts of its experience. Cellular clocks continue turning through the night.

How much of that story could we tell with a single probability?

Epistemic Physics invites us to travel with a larger notebook. Keep a direction and an amount together. Keep the observations that contributed, the contexts in which they mattered, and the clock that governs their influence. A present answer can hide very different machinery.

This is our guided tour of neuroscience. Each stop opens into a chapter with its own experiments. Computational neuroscience is the approach of using mathematical models to understand nervous systems; it travels through all seven stops, from individual cells to whole animals.

Pack a balance and an amount

Start with a binary question: does a cue point toward food? Let $Y$ be supporting trial amount and $N$ opposing trial amount. A trial is one modeled observation; a tracelet is a contribution carried into an evidence description. Lineage records which observation a contribution came from.

With a symmetric half-unit baseline on each side, define

The total $A$ includes that baseline, called a prior. The balance $P$ subtracts opposition from support. A Beta distribution with positive shape parameters $\alpha ,\beta$ has mean

Here $\mu$ is a mean probability and $v$ its centered version, between $-1$ and $1$. The coordinates satisfy , placing the state inside the evidence cone. For this Beta model, the probability variance is $\mu (1-\mu )/(A+1)$.

Both a quiet learner and one with many trials on each side can report $\mu =1/2$. Their next observations have very different effects. Malleability means how readily a belief changes; amount and the update rule help determine it.

Now try the same bookkeeping habit in a cell, a population, and a moving body. We will give each model its own positive weights and units. The connection becomes visible when we line up the quantities:

Read across a row, then down a column. The common weighted-average structure gives us a way to move between these worlds while keeping their meanings legible.

1. Enter a cell through its open doors

A neuron’s membrane contains channels through which electrical current flows. A channel’s conductance measures how easily current flows; its reversal potential is the voltage toward which it pulls the membrane.

In the passive model, excitatory, inhibitory, and leak channels combine into

The $g$ values are conductances and the $E$ values reversal potentials. The subscripts identify excitation, inhibition, and leak. This is the familiar weighted average from Neuronal Dynamics, §13.6.3.

Multiplying all conductances by the same positive factor leaves the equilibrium voltage $V_{\mathrm{\infty }}$ unchanged. Yet a sustained current $I$ displaces the voltage by $I/G$, and capacitance $C$, the membrane’s ability to store charge, gives a settling time $C/G$. More open doors produce a smaller displacement and faster settling.

Predict before pressing a button: can a cell be quick to settle and difficult to push far? Try two cells with the same voltage.

The cellular and synaptic chapter opens the exact coordinate change, including a leak that pulls toward an off-center voltage. It also follows a surprising first instant: equal capacitances give the same initial pulse slope $I/C$, even when the later trajectories separate.

2. Let the animal move its question

A sensor receives different information in different places. Two objects can give the same signal at one position and different signals a short distance away. Movement changes the question the sensor asks.

Weakly electric fish provide a vivid example. Stamper and colleagues found increased whole-body oscillations and tail bending under degraded sensory conditions. Behavior and sensing form a loop: what the animal knows influences how it moves, and how it moves influences what it can learn.

The sensory neuroscience and neuroethology chapter gives you a movable sensor and a budget of four readings. Compare staying still, visiting one side, and sampling both sides. Then introduce a calibration offset shared by the readings. A good path can help separate a feature of the world from a feature of the instrument.

The same chapter asks how several senses should combine their clues. In Ernst and Banks’s visual-haptic experiment, people combined seeing and touching in a way consistent with giving more weight to the more precise estimate. Precision here is inverse modeled variance. That positive-weight average brings us back to the notebook, with an observation model explaining what the weights mean.

3. Widen the window to a population

Imagine neurons arranged around a circle, each preferring a different heading. A bright patch of activity can travel around that ring as an animal turns. Experiments by Kim and colleagues investigated persistent heading patterns in the fly brain and their responses to imposed activity.

For a chosen nonnegative activity weight $w_i$ and preferred angle ${\varphi }_i$, define

The bold symbol $\mathbf{P}$ is a two-component vector. The normalized arrow $\mathbf{P}/A$ summarizes the pattern when .

But two equally strong opposite patches cancel their arrows. So does activity spread evenly around the ring. Both can have the same total $A$. What would a navigator lose by carrying only that summary?

The systems neuroscience chapter puts the complete population beside its compass arrow and an average of doubled angles. That extra view reveals a pair of opposite possibilities. It also asks how recurrent connections, which send activity back through the network, could sustain a heading through time.

Time deserves its own window. A sampled probability held for several moments makes neighboring outcomes share an internal state. The same mean response can accompany very different count variances:

The experiment specifies exactly when a state is redrawn. That clock is as much a part of the model as its distribution.

4. Let memory decide how much the present can change

Return to the fish’s food cue. Yesterday’s world may be useful today, or it may have changed.

For the binary Beta model, adding one outcome $x\in \{0,1\}$ updates the mean exactly as

The fraction $1/(A+1)$ is the learning gain: the portion of the gap to the outcome closed by this trial. Large accumulated amount makes that step small.

Now choose a memory rule: multiply old raw amounts $Y,N$ by a retention fraction $\lambda$, between zero and one, before adding new input. Keep the prior in place. The learner can give recent observations more influence and remain more malleable in a changing world.

Synaptic plasticity means changes in the effectiveness of connections between neurons. Aitchison and colleagues explore a model in which synapses track uncertainty and use it to adjust learning. Our two-pool learner offers a small, transparent doorway into that idea.

The learning, memory, and plasticity chapter continues into contexts and sleep replay. A memory can be practiced repeatedly while its original observation count stays fixed. In the demo, rehearsal improves a circuit’s approximation to what its stored observations already say. New observations, better use of old observations, and forgetting become three different operations you can inspect.

5. Build a machine from the same ingredients

A neuron-like model can keep supporting drive $T$ and opposing drive $F$ as separate nonnegative pools. With baseline , its centered mean is

The numerator sets the direction; the denominator regulates its scale. In neuroscience, divisive normalization describes computations in which a response is divided by a quantity involving pooled activity. Heeger’s normalization model offers a classic example. The shared denominator structure invites us to compare how circuits retain or discard their total drive.

The neural networks and computation chapter develops the β-Vale family. One version reads a deterministic value. Another samples an internal state. An episodic version computes in steps; a continuing version lets spikes arrive over time. Sampling is a choice within the family, and each architecture supplies its own timing and connection rules.

A useful design question follows: could a network preserve the amount it normalized away and use it later to control learning, sampling, or the next measurement? The demos let the same internal state try several jobs.

6. Listen for the clock behind the chorus

Eight musicians play equally loudly. If their oscillating signals line up, the chorus has a large peak. Spread them evenly through a cycle and their oscillating parts can cancel.

A phase is a position within a repeating cycle. Cellular clocks can keep strong individual rhythms while the measured ensemble rhythm becomes weak. Hatori and colleagues studied a regulator involved in maintaining synchrony among clock neurons in the brain’s suprachiasmatic nucleus, the SCN.

The rhythms and biological clocks chapter puts phase arrows on the same circle used for headings. Move a clock hand, couple the clocks, or deliver a light pulse. Compare the individual rhythms with the chorus they create.

It then descends another level: what makes an individual clock? A fading memory can follow an external rhythm. Delayed negative feedback, a correction based on an earlier state, can destabilize a steady state; a suitable nonlinear rule can sustain a bounded oscillation. These are different mechanisms to build and explore. A phase in a day and accumulated epistemic exposure keep their separate jobs.

7. Put the notebook into a moving body

Hold a cup lightly, then hold it firmly without moving it. The position stays similar while resistance to a disturbance changes.

Opposing muscles can act together, a behavior called co-contraction. In a two-spring model, their stiffnesses determine both an equilibrium position and the resistance to a push. Scale both stiffnesses together and the equilibrium stays fixed while the total stiffness grows. Compare this with scaling the membrane’s conductances, or increasing Beta amount at a fixed mean.

Burdet and colleagues investigated how people learned arm impedance to stabilize movements in an unstable environment. Impedance describes how force and motion relate, including stiffness, damping, and inertia.

The neuroengineering and motor-control chapter starts with the opposing springs, then follows estimation and feedback into brain-computer interfaces. Drag a target, stiffen the model arm, and compare the same external push. Next, try a tracker while sensory readings disappear. The estimate, the body’s resistance, and the policy choosing an action become separate parts of one loop.

An animal-sized hypothesis

Let’s assemble an adventurous model animal. Give it a directional sensory ledger, a source-aware memory, a population that preserves several possible headings, and a body with adjustable stiffness. Let its daily phase help choose when to explore, consolidate, or act.

Three experiments could begin to connect the pieces:

• Spend a fixed sensing budget. Let the animal choose movements that distinguish its remaining hypotheses. Compare prediction error after equal numbers of new trials, including shared sensor offsets.
• Separate experience from rehearsal. Keep the same observations and computational budget, then compare direct recall with context-sensitive practice. Ask whether improved behavior comes from new information, improved computation, or both.
• Coordinate learning and action. Let retained trial amount influence learning gain, while predicted disturbances influence bodily stiffness. Compare this coupled policy with independent controls under sudden environmental changes.

A larger possibility is a nervous system that manages several kinds of resistance: electrical response, resistance to revising a prediction, persistence of a population pattern, and mechanical stiffness. The equations above give each kind a concrete meaning. The adventure is to discover which couplings make a more capable animal.