β-Vale networks: one state, many ways to compute

Miraia S. Chiou

Survey · Neuroscience

β-Vale networks: one state, many ways to compute

Explore two pools of neural drive through static, episodic, and spiking networks.

2026-09-08 · 14 min read · 3 interactive demos

Imagine a neuron with two baskets: one catches signals that support its activity; the other catches signals that oppose it. Their balance suggests a response. Their combined amount tells us something about how firmly that balance is held.

Now give this neuron a few choices. Should it produce one predictable answer, or draw a possible answer from a distribution? Should it calculate once, work through a short episode, or keep responding as events arrive? Should its neighbors receive activity levels or individual spikes?

These choices make β-Vale networks, or Beta Valence Networks, a flexible family. Developed by Miraia Chiou, they share a two-pool representation while leaving considerable freedom in how a network computes.

Design choice	Some possibilities
Readout	A deterministic function of the state, or a sample from its Beta distribution.
Memory and clock	A fresh static evaluation, a bounded episode, or continuing dynamics.
Communication	Real-valued activity levels, or discrete spikes.
Connections	Forward pathways, recurrent loops, or combinations of both.

These choices can be combined. A static network can sample. A spiking network can be entirely deterministic. A recurrent network can run for a short episode and then clear its state.

Let’s start with the simplest version and add ingredients one at a time.

Two pools, several possible readouts

Call the supporting drive $T$ and the opposing drive $F$ . Both are nonnegative. Their total is $n = T + F$ .

In a static network, incoming activity levels and connection weights determine these pools for the current input. An activity level can represent a firing rate, meaning spikes per unit time, or simply a numerical feature passed between artificial neurons. The network then evaluates its output without carrying a changing state from one example to the next.

The two pools also describe a Beta distribution:

Beta (T + c, F + c) .

The positive constant $c$ supplies a balanced prior, the baseline before incoming drive arrives. Two useful choices are $c = 1$ , which gives a uniform baseline, and $c = \frac{1}{2}$ , the Jeffreys prior, whose baseline density rises toward both endpoints.

We can use the distribution as a portrait of the state while computing a completely deterministic output. Its mean, written $μ$ , and centered mean, written $v$ , are

μ = \frac{T + c}{T + F + 2 c}, v = 2 μ - 1 = \frac{T - F}{T + F + 2 c} .

$μ$ lies between zero and one. $v$ expresses the balance on a scale from $- 1$ to $1$ . For example, a mean of $0.8$ becomes a centered mean of $0.6$ .

Two deterministic firing rules follow naturally:

\begin{aligned} full range: & r = r_{max} μ, \\ positive centered part: & r = r_{max} max (0, v) . \end{aligned}

Here $r$ is the output rate, $r_{max}$ is its maximum, and $max$ selects the larger of its arguments. The second rule makes an opposing or perfectly balanced state silent. Choosing a response rule is part of designing the neuron.

Other deterministic versions can read the raw difference $T - F$ , a normalized difference $(T - F) / (κ + T + F)$ with positive scale $κ$ , or a density mode when a suitable mode exists. Each choice emphasizes a different aspect of the same two pools.

Mean, mode, and amount answer different questions

The mean is the average of the distribution. A mode is a location where its density peaks. The amount $n$ is the total incoming drive. None of these is interchangeable with the others.

With the uniform prior $c = 1$ and positive $T, F$ , the centered mode is $(T - F) / (T + F)$ . Other priors change the mode formula, and some Beta distributions have endpoint peaks, two modes, or no unique mode. The response-cloud demo below labels the actual modes separately from the mean.

The pools can represent effective neural drive. Reading them as literal trials additionally requires a unit of evidence and a lineage policy: an account of where the inputs came from and when they repeat the same underlying source.

A familiar neural computation: divide by the crowd

Imagine one violin playing into a microphone. Its signal matters, but so does the orchestra around it. A neural response can similarly depend on both its own drive and the activity pooled around it. Divisive normalization expresses this idea by putting pooled activity in a denominator.

In Heeger’s 1992 model of cat visual cortex, interactions among modeled cells normalize responses with respect to stimulus contrast, helping account for physiological response patterns. For our comparison, collect nonnegative processed drives into variables $x_{j}$ and use this simplified normalization form:

\begin{aligned} normalization: & r_{i} = \frac{x_{i}}{κ_{N} + \sum_{j} x_{j}}, \\ opponent: & v = \frac{T - F}{κ_{B} + T + F} . \end{aligned}

The index $i$ identifies a cell; $j$ runs over its normalization pool. The quantity $r_{i}$ is a dimensionless response fraction, which a rate scale can convert to firing rate. The positive baseline $κ_{N}$ sets the drive scale at which pooled activity substantially affects the denominator. Actual normalization models can specify powers, pooling weights and additional response transformations. This compact expression isolates the shared operation.

In the β-Vale expression, $T$ and $F$ are our two nonnegative drives. The baseline is $κ_{B} = 2 c$ when $v$ is the centered mean of $Beta (T + c, F + c)$ . The two kappas play analogous denominator roles; their numerical values and units belong to their respective models.

The denominator has the same structure: baseline plus pooled nonnegative drive. The β-Vale numerator adds an opponent comparison, subtracting one pool from the other. We can make that connection exact with two normalized channels:

q_{T} = \frac{T}{κ_{B} + T + F}, q_{F} = \frac{F}{κ_{B} + T + F}, v = q_{T} - q_{F} .

Both channels stay nonnegative even when their difference is negative. Their sum is less than one because the baseline occupies part of the denominator. Adding $c$ to each numerator gives the complementary Beta mean channels, whose difference is still $v$ . A firing rule can subsequently keep only the positive part, as above.

Predict before calculating: if both drives double, does the centered response double? With $T = 3$ , $F = 1$ and $κ_{B} = 2$ , the response is $1 / 3$ . Doubling the drives gives $4 / 10 = 0.4$ . More drive weakens the baseline’s influence while preserving the ratio of the incoming pools.

More generally, multiplying both pools by $a > 0$ gives

v (a) = \frac{a (T - F)}{κ_{B} + a (T + F)} ⟶ \frac{T - F}{T + F} as a ⟶ \infty,

provided $T + F > 0$ . Exact common-scale invariance holds with zero baseline; a positive baseline makes it an asymptotic property. Equal opposing pools remain balanced at every scale, while their total still changes the effect of the next small input.

This makes normalization a useful bridge between neural computation and the two-pool state. Interpreting the drives as independently sourced trials additionally requires the observation and lineage model described above. For a physical weighted denominator, visit the membrane’s open channels. To explore what retained inputs and repeated experiences contribute over time, continue to what a sleeping brain can rehearse.

The same circuit can be deterministic or stochastic

An opposing neuron still emits a nonnegative activity level. Its output polarity determines where that activity goes: a supporting cell contributes to the next cell’s $T$ pool, and an opposing cell contributes to its $F$ pool. Keeping one outgoing polarity per cell is a Dale-style wiring rule.

The little network below has two sensory inputs, a supporting hidden cell, an opposing hidden cell, and an output cell. A hidden cell is simply an intermediate neuron between the inputs and output. The connection weights are fixed.

Start by propagating means. Then select a stochastic pass, which draws an activity from each cell’s Beta distribution. The wiring and readout scale stay the same; the choice of representative value changes.

Positive rates, opposite effects

One circuit. Two readout choices.

Drag two sensory inputs, then follow a forward pass. The teal cell E supports the output; the rose cell I opposes it. Both fire at nonnegative rates. Begin with deterministic means, then try sampled activity. The histogram below shows the stochastic version for comparison.

Supporting routeOpposing routeOutput

Sample averageMeans propagated

Mean-path output: 0.640
Average of 512 sampled outputs: 0.639
Means propagated: 0.640
Stochastic output SD: 0.189

This pass uses means throughout; changing the sample batch leaves its answer unchanged. A large inhibitory rate still has a positive value; its rose connection sends it into the output's opposing pool. Hidden samples also change the output's Beta shape, so the output cloud combines many conditional Beta distributions.

An untrained illustrative circuit, with fixed routes, nonnegative weights and a Jeffreys prior at each receiving neuron. The sensory channels are x, 1−x, y and 1−y. E favors x; I favors y. The output receives T = 6r_E and F = 6r_I, with rates normalized to [0,1]. SD means standard deviation. Both readout choices evaluate a static forward pass without retained memory.

Open the circuit in the playground.

A deterministic pass gives the same answer whenever its inputs are the same. In the stochastic version, hidden samples change the output neuron’s two pools. Across many passes, the output therefore combines many conditional Beta distributions into a mixture. Calling each local state a Beta does not make the whole output cloud one Beta.

What optional sampling adds

To sample, draw a value $B$ between zero and one:

B \sim Beta (T + c, F + c) .

The symbol $\sim$ means “is drawn from.” We can replace $μ$ by $B$ in the full-range rule, or replace $v$ by the centered sample $S = 2 B - 1$ in the thresholded rule.

The distribution’s spread now directly shapes the response cloud. Try little balanced drive and much balanced drive. Both have mean $1 / 2$ , but they offer very different bags of possible responses.

Optional sampling in a β-Vale neuron

One state, a cloud of responses.

Try little input and strong balanced input. Both can have a 50% mean, yet their samples look very different. Then switch on a firing threshold: averaging before the threshold can give a different answer from averaging after it.

Supporting drive T 2 Opposing drive F 2

Beta density and samplesMean / firing at the meanMode or modesAverage firing

Beta mean: 50.0%
Beta mode: 50.0%
Mean firing E[r]: 50.00 Hz
Firing at mean r(E[B]): 50.00 Hz
Rate SD: 20.41 Hz
Sampled mean firing: 48.79 Hz

The full-range rule is linear, so firing at the mean equals mean firing. Every firing rate remains nonnegative.

B ~ Beta(2.5, 2.5); maximum rate 100 Hz is illustrative. SD means standard deviation. These are independent static passes, not a spike train. Uniform density has no unique mode.

Open the response cloud in the playground.

Sampling can also change average activity. With a uniform $B$ , the mean is $1 / 2$ . The zero-threshold rule applied to that mean gives silence. But half the individual samples lie above the threshold, and their contributions give an average rate of $r_{max} / 4$ .

This is nonlinear averaging: averaging after a nonlinear transformation can differ from transforming the average. With full-range firing, which is linear, the two calculations agree.

A small appearance of π

With no incoming drive and a Jeffreys prior, $B \sim Beta (\frac{1}{2}, \frac{1}{2})$ . For the zero-threshold rule,

\frac{E [r]}{r_{max}} = \int_{1 / 2}^{1} \frac{2 b - 1}{π \sqrt{b (1 - b)}} d b = \frac{1}{π} .

$E [r]$ means the expected, or long-run average, rate. It is about 31.8% of the maximum in this particular construction. Changing the prior or response rule changes the result.

Three ways to give the network time

Picture three desks receiving the same messages. At the first, each message gets a fresh calculation. At the second, someone keeps notes while solving one problem and clears the desk for the next. At the third, the notes remain, with old entries gradually fading.

Those desks suggest three versions of a β-Vale network:

Static: compute from the present input. A layered forward pass is enough; evolving memory is optional.
Episodic: give one example a bounded sequence of internal steps. Signals can circulate or become spikes during this short computation, then the retained state is cleared before the next example. Here “episodic” means a computation with a beginning and end, rather than autobiographical memory.
Continuing dynamics: let the state evolve as inputs arrive over time. Old drive may decay, new drive enters, and the network carries its recent history forward.

The episodic version provides a useful middle ground. A still image, for example, could receive several internal processing steps without requiring an indefinitely running simulation. These steps form a microtime, an internal computational clock whose steps need not equal biological milliseconds.

A fully spiking dynamical version goes further: incoming spikes update the two pools, the state controls outgoing spikes, and those spikes become events for downstream cells. Connection delays specify when they arrive. Recurrent connections, which loop activity back into the network, can be added under either an episodic or continuing clock.

Continuous-valued activity levels can also evolve in time. Spiking is one communication choice within a dynamical network.

A spike can leave the neuron’s state intact

How can a deterministic state emit separate spikes? Add a small bucket called communication charge, written $z$ . At each step, a nonnegative activity fraction fills it. When it reaches one, emit a spike and subtract one from the charge.

The neuron now has two kinds of memory: the $T, F$ pools hold accumulated drive, and $z$ tracks progress toward the next outgoing event. Emptying some charge after a spike can leave both drive pools intact. Clearing an episode can reset all three.

This threshold-and-reset idea is also familiar from integrate-and-fire neurons. It supplies a timing mechanism in addition to the Beta state. Random draws are optional.

Try the three clocks below. First send a short pulse and watch what remains after it stops. Then send strong balanced input: can a silent neuron still hold a large amount of drive?

Pulse composer

Two inputs, a burst of spikes

Place two support pulses in twelve slots. Can timing alone produce six output spikes before the episode ends?

A small input budget, a lasting response

Tap a slot to move pulse 1. Each pulse carries one support input.

Predict how many spikes your timing will produce, then run it. The goal is an optional invitation; every arrangement is available.

Run other inputs, sample the Beta, and inspect all three clocks

Step 12 of 48 · No random draws

Supporting input TOpposing input FCurve: activity fractionLower ticks: output spikes

Static

Supporting T: 1.00
Opposing F: 0.00
Activity fraction: 0.500

Episodic

Supporting T: 12.00
Opposing F: 0.00
Activity fraction: 0.923
Charge left: 0.820
Total spikes so far: 9

Continuing

Supporting T: 7.79
Opposing F: 0.00
Activity fraction: 0.886
Charge left: 0.522
Total spikes so far: 9

Both spiking cells are fully deterministic: activity fills a charge bucket, and reaching one emits a spike. Randomness is a separate choice.

An illustrative receiving cell with unit-weight incoming events and a Jeffreys prior of ½ per side. Activity is max(0, 2B−1), using the mean or a draw for B. The spiking cells add that activity to charge; a spike subtracts one from charge and leaves T and F intact. Each episode clears T, F and charge before its first input. The continuing cell discounts its drive pools with an eight-step half-life; its prior stays fixed. If a continuing step represents 10 ms, that half-life is 80 ms. The episodic steps are computational steps without a physical time calibration. Spike totals count the whole displayed run across episodes.

Open the clock comparison in the playground.

The three rows use one illustrative set of rules. An episodic network could also forget within an episode; a continuing network could retain some pools indefinitely. The clock, forgetting rule and spike generator can be chosen separately.

One explicit update rule for the spiking rows

Let $I_{t}^{T}$ and $I_{t}^{F}$ be the total weighted supporting and opposing input spikes arriving at step $t$ . A simple drive update is

T_{t + 1} = λ T_{t} + I_{t}^{T}, F_{t + 1} = λ F_{t} + I_{t}^{F} .

$λ$ is the fraction retained each step. The demo uses $λ = 1$ within an episode and $λ = 2^{- 1 / 8}$ for its continuing cell, so retained drive halves in eight quiet steps. Discounted pools are effective amounts of retained input. The prior stays separate and is added once when forming the Beta state.

From the updated state, obtain an activity fraction $u_{t}$ between zero and one using the positive centered mean or sample. Then

\begin{aligned} {\tilde{z}}_{t + 1} & = z_{t} + u_{t}, \\ s_{t} & = 1 [{\tilde{z}}_{t + 1} \geq 1], \\ z_{t + 1} & = {\tilde{z}}_{t + 1} - s_{t} . \end{aligned}

The indicator $1$ equals one when its condition is true and zero otherwise. Thus $s_{t}$ records an output spike. With this bounded activity and initial charge zero, there is at most one output spike per step. An episode boundary clears $T, F, z$ before its next input; a spike only subtracts from $z$ .

Turning this into a physical-time model also requires specifying the duration of a step, incoming delays, and whether charge leaks or a refractory period temporarily prevents another spike. The demo makes a particular choice: charge does not leak, and it adds no refractory period. Other choices create other members of the family.

The geometry travels with the state

The connection to Epistemic Physics remains available across these versions. Write

α = T + c, β = F + c, A = α + β, v = \frac{α - β}{A} .

$A$ is the total including the prior, and $v$ is the centered mean. These are the total-amount and velocity coordinates of the Beta state. With $c = \frac{1}{2}$ , $A$ is the usual epistemic time coordinate $τ_{E} = T + F + 1$ .

Two cells can share the same balance but carry different amounts. If we add the same small drive to each, the cell with more existing drive changes its balance less. This is a local sense of malleability, the ease of changing a state. A static cell can have this response sensitivity even though it retains no memory between evaluations.

If we choose to sample, that same geometry determines the spread:

Var (S) = \frac{1 - v^{2}}{A + 1} .

$Var$ denotes variance, the average squared distance from the mean. At fixed $v$ , more total amount gives a narrower response cloud. When we use deterministic readouts, the Beta variance still describes the associated distribution, but repeated identical evaluations produce the same output.

Where epistemic mass appears

The mass coordinate is $m_{E} = 2 \sqrt{α β}$ . Since $m_{E}^{2} = A^{2} (1 - v^{2})$ ,

Var (S) = \frac{m_{E}^{2}}{A^{2} (A + 1)} .

This is an exact identity for the local Beta distribution. Holding $A$ fixed while changing the balance is a different comparison from increasing $A$ at fixed mean.

A family with room to explore

There is freedom in the wiring too. We can use sparse connections, combine direct and processed pathways, or give a neuron several dendritic subunits, local input-processing branches whose outputs are combined. Different cells could have different priors, thresholds, forgetting times and outgoing polarities.

Learning adds another choice: gradient-based training can adjust weights to reduce an error, while local rules could adjust a connection from nearby activity or spike timing. A particular state representation leaves room for several learning methods; their usefulness has to be explored in actual tasks.

Some especially interesting comparisons sit close together: a static deterministic network and its sampled version; a short episode and a longer one; a rate-based circuit and a spiking circuit with comparable inputs. We can ask which changes improve decisions, which preserve useful history, and which merely add variability.

The common thread is small enough to keep visible throughout: two nonnegative pools, their balance and amount, a readout, and a policy for carrying state forward.

For the broader neuroscience connections, return to A brain in the evidence cone.

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