A talk I gave this week. A visual object is a computer science metaphor for a representation of a useful visual regularity.  This talk discusses how auditory objects are constructed, and how the acoustic sensory apparatus differs from vision.

Auditory Object Talk

Until next time.

Part Of: Logic sequence
Content Summary: 700 words, 7 min read

Organizational Principles Of Logic

The ingredients of any system of logic are:

• A proposition is an atomic statement that can acquire a truth value. For example, “Socrates is a man”.
• A connective takes atomic propositions, and melds them into a more complex, composite proposition. For example, AND is a connective.

Propositions, in both atomic and composite form, represent containers for truth valuations. A judgment is a filled container: a proposition assigned a specific truth value (true or false).

But logical systems are not static entities. The heart of any logic is its dynamics: rules which permute its resident propositions.

The IPL System

Once upon a time, logicians tired of the tedious algebraic format of their logical systems. Natural deduction was invented as a graphical alternative to such systems.

In this post, I use the natural deduction format to present one particular system of logic, Intuitionistic Propositional Logic (IPL).

IPL permits the following connectives:

• ∧ (conjunction, “AND”)
• ⊃ (implication)
• ∨ (disjunction, “OR”)
• ⊤ (truth)
• ⊥ (falsity)

Rules can be categorized as follows:

1. Introduction rules show how connectives are injected into the system.
2. Elimination rules describe how connectives are removed from the system.

Ready to take a look under the hood?

Conjunction (∧) Rules

The ∧ (AND) connective features three rules. In the following visuals, premises (knowledge before rule application) are on top, conclusions (knowledge after rule application) below.

Our first rule is AND-Introduction (∧I). It permits us to glue facts together.

In English: “Before we applied the ∧I rule, suppose we know two facts: A, and B. Afterwards, we know one additional fact: A∧B.”

Next, we have Left-AND-Elimination (∧EL) and Right-AND-Elimination (∧ER). Together, these rules “remove the glue”.

The left rule means: “If we use A∧B, we can use A by itself.”

Implication (⊃) Rules

IMPLICATION-Elimination (⊃E) proceeds fairly intuitively:

IMPLICATION-Introduction (⊃I) is where things get tricky. Consider the weaker version of this rule, on the left.

The ellipsis “..” means that other rules may be injected, the intermediate proof can be hundreds of lines long, if need be. So this weak rule seems intuitive:

If you know A, and from this information can eventually prove B, then you may conclude ‘A implies B’.

However, this version is too simplistic: it allows implications only to be introduced when the antecedent was already assumed. The real rule is more powerful:

If you assume A, and from this information can eventually prove B, then you may conclude ‘A implies B’.

Get the difference? Before, our assumptions were unchanged.  But now we expand our assumption set, and denote the location of our assumption-expansion with a line, and name it (in this case, “a”.)

Disjunction (∨) Rules

OR statements only require only one of its terms to be true. If you have evidence for one statement, it doesn’t matter whether the others are true or false.

This intuition is cashed out in the Left-OR-Introduction (∨IL) and Right-OR-Introduction (∨IR) rules.

The left rule means: “If we use A, we can use A∨B.” 

Of all the rules in Intuitionistic Propositional Logic, OR-Elimination (∨E) requires the most time to comprehend. 

Notice that the last two inputs fit the criteria for implication, and could simplify to A⊃C and B⊃C.

Here is one way to interpret the above rule:

Suppose we know that A∨B.

If it is true that “A implies C” and “B implies C”, then we know C is true.

We know this because at least one component of A∨B must be true!

Truth (⊤) and Falsity (⊥) Rules

IPL includes two rules regarding Truth and Falsity.

The TRUTH-Introduction (⊤I) rule simply means that the system admits trivial truths.  There is no TRUTH-Elimination rule.

The FALSITY-Elimination (⊥E) rule reflects the Principle Of Explosion: “from contradiction, anything follows.”

Notice that the IPL System connective for complement (¬). However, Falsity allows the system to express negation regardless: ¬A = A ⊃ ⊥.

Takeaways

In this post, we learned the ten rules which define Intuitionistic Propositional Logic (IPL):

Next time, we’ll use IPL to solve a real problem. 🙂

I made a graphic explaining my current macro-theory of emotion.

 

Enjoy!

Part Of: Neuroeconomics sequence
Content Summary: 1000 words, 5 min read

Reentrant Loops

The thalamus receives input from the entire cortex. The cortex & thalamus like two nested spheres, innervating one another. Let us call such cortical-subcortical loops, reentrant loops.

The basal ganglia also participates in a reentrant loop. The basal ganglia and thalamus are nestled inside one another, far below the cortical mantle:

The last major reentrant loop involves the cerebellum. Taken together, reentrant loop anatomy looks something like this:

Dopamine Anatomy

We turn now to the basal ganglia. The basal ganglia is innervated with dopamine, and contains an order of magnitude more dopamine receptors than any other brain region (Dawson et al, 1986). To understand this neurotransmitter, we must consider its biochemical signature.

As a molecule, dopamine contains catechol (a kind of benzene C6H6) and a side-chain amine. Thus, it participates in the catecholamine family.  

There are in fact many kinds of catecholamines. For example, from phenethylamine you can synthesize tyrosine (via hydroxylation). In fact, there are thirteen different catecholamines, with a particular tree of biosynthetic sequences:

Three particular catecholamines are major players in brain neurochemistry.  

1. Epinephrine (EPI), i.e. adrenaline, promotes the fight-or-flight response.
2. Norepinephrine (NE) encodes perceptual salience.
3. Dopamine (DA) is the currency of motivation and reward.

Thus, catecholamines play a role in metabolic, perceptual, and motor arousal. In short, they are arousal systems.

Within the brain, a small number of discrete cell groups manufacture dopamine:

In the above cell groups, two dopaminergic cell groups are particularly important: A9 (SNc) and A10 (VTA). The nigrostriatal pathway emerges from the former, and the mesolimbic and mesocortical pathways emerge from the latter.

Basal Ganglia Anatomy

The basal ganglia is an ancient structure, with four main components:

1. The primary input structure, the Striatum
2. Global Pallidus External (GPe)
3. Subthalamic Nucleus (STN); and
4. three interrelated structures: the Global Pallidus Internal (GPi), Substantia Nigra Pars Reticulum (SNr), and Ventral Pallidum (VP)

These components comprise the core of the CBTC Loop:

In the above, basal ganglia nodes are in purple. Black arrows are excitatory, red are inhibitory, green are dopaminergic.

Three Circuits: Direct, Indirect, Hyperdirect

Can you find the loop in this circuit? In fact, if you look closely, there are three:

Consider the direct pathway. We see that the thalamus has an excitatory influence on the cortex. However, the GPi/SNr/VP node inhibits the thalamus. However, if an activated striatum inhibits the GPi/SNr/VP, the overall effect is “releasing the brakes”. The direct pathway is excitatory.

In the indirect pathway, the STN buffers the GPi/Snr/VP “stop” signals, and attempts to “release the brakes” from the GPe is inhibited by the Striatum.  Three inhibitory tracts along the indirect pathway are net-negative. Thus, the indirect pathway is inhibitive.

The hyperdirect pathway has one inhibitory track (the GPi/SNr/VP “brakes”) – thus this pathway inhibits action. Why did evolution select two inhibitory pathways?  One answer might rely on circuit length.  The hyperdirect pathway bypasses several stages of the basal ganglia: perhaps its use was to accelerate the inhibitory signal of the indirect loop. 

The most prominent cell type in the striatum is the medium spiny neuron (MSN). MSNs express one of two kinds of dopamine receptors: D1 and D2. Cells expressing these receptors are thoroughly mixed in the striatum. However, these pathways are strictly segregated by dopamine receptor:

• D1-expressing MSNs only participate in the direct pathway, and
• D2-expressing MSNs only participate in the indirect pathway.

Basal Ganglia As Action Selector

Simple organisms act in accordance to simple stimulus-response pairings, by innate fixed action patterns. More complex organisms, in contrast, actively choose between multiple different response candidates. Such an organism would need a selector which makes this decision. To be effective, a selector must express the following properties

1. Singleton. If there are incompatible options, the selector must choose a single behavior. Once a behavior has been activated, the selector must also actively inhibit other behaviors, to promote unanimity of purpose.
2. Reward Maximization. The selector must choose the best option, comparing their merits in a common currency.

Redgrave et al (1999) argue that the basal ganglia is a selector, that is has the structural and functional properties that we would expect from such an architecture. The direct pathway disinhibits the winner, the indirect pathway inhibits the losers.

References

• Joel & Weiner (1997). The connections of the primate subthalamic nucleus: indirect pathways and the open-interconnected scheme of basal ganglia-thalamocortical circuitry.
• Nougaret et al (2013). First evidence of a hyperdirect prefrontal pathway in the primate: precise organization for new insights on subthalamic nucleus functions.
• Redgrave et al (1999). The basal ganglia: a vertebrate solution to the selection problem?

Part Of: [Neuroanatomy] sequence

Table Of Contents

• Circulatory System and BBB
• Ventricular System and BCSFB
• Endocrine System and CVOs
• Visceral vs Cognitive Processing
• The Folly Of Spock

Circulatory System and BBB

Cells need blood to survive. Neurons are cells like any other. Thus, we expect the circulatory system to extend into the brain.

Blood flow in the brain is organized around a structure poetically named the Circle Of Willis.

Unfortunately, capillaries sometimes leak, releasing noxious bacteria. Such infections are usually addressed by your immune system. However, certain areas of the body cannot tolerate such risks. If you had to choose, which areas of the body which you want special protection from infection?

Natural selection has produced specialized protection for three areas of the body: the brain, the eyes, the gonads.  Damage to these structures are particularly destructive to an organism’s fitness. Capillaries in these “protected zones” are given essentially several extra layers of “armor”, which prevent bacteria from escaping.

This protective mechanism is known as the blood-brain barrier (BBB). This armor is metabolically expensive, which explains the absence of “blood-foot barriers”.

Ventricular System and BCSFB

Your brain does not rest against the base of your skull. That would destroy brain tissue. Instead, it is immersed in a fluid bath, which protects and supports the brain.  Formally, the fluid is known as cerebrospinal fluid (CSF), and it is found as a support for the brain (ventricular system) and surrounding/cushioning the brain (in the subarachnoid space).

Cerebrospinal fluid (CSF) takes no direct connection to the circulatory system. In fact, since the CSF directly contacts nervous tissue, it also requires extra protection from infection. The blood-cerebrospinal fluid barrier (BCSFB) exists for precisely this reason.

Taken together, the BBB and the BCSFB insulate the brain from the circulatory system, while still allowing nutrients to reach neurons:

Endocrine System and CVOs

In Towards Body Architecture we explored how the nervous system interacts with other bodily systems (respiratory, digestive, etc).  In fact, the nervous system acts as a control system, influencing how other systems perform. No other anatomical system can claim that…

…except one. Your endocrine system also acts as a control system! Hormones play a role in digestion, cell growth, reproduction, etc – a whole host of bodily processes. Speaking generally, nervous signals act quickly (nerve signals do their work within seconds), whereas endocrine control (hormone signals enact change more slowly, usually in minutes or hours).

Two independent control systems is a poor way to design a body. We have every reason to benefit from them synchronizing their efforts, exchanging information. Let’s examine stress as an example, which requires the involvement of both systems.

• The nervous system is responsible for detecting stressful stimuli. It then transmits a signal, via nerves of the sympathetic nervous system to the adrenal gland, which immediately begins manufacturing cortisol (a hormone which mediates stress). This is fight-or-flight, a fast response.
• The endocrine system concurrently releases adrenocorticotropic hormone (ACTH) which also prompts the adrenal gland to manufacture cortisol. However, ACTH takes longer to take effect, and stays in your system longer. Thus, the endocrine system supports non-transient forms of stress.

These two phenomena are obviously correlated. But how? Your endocrine system is confined to your circulatory system – hormones travel via the bloodstream. But we just learned that the brain is insulated from the bloodstream…

The nervous system and the endocrine system coordinate with one another via circumventricular organs (CVOs). CVOs puncture the blood-brain barrier in a controlled way. They come in two flavors:

• Sensory CVOs translate hormone messengers into neural signals.
• Secretory CVOs translate neural signals into hormone messengers.

As you might expect by now, these two categories of CVOs provide neuroendocrine integration.

Visceral vs Cognitive Processing

In my last post, I presented the cognitive perception-action cycle. Let us consider these cybernetic systems side by side.

This image situates visceral and cognitive processes together. This reflects the medial viscera principle:

Visceral processes tend to reside in the center of the brain (medial regions).

The neuroendocrine system does not operate alone. Rather, it works alongside the autonomic nervous system, which regulates the body via two complementary systems:

1. Sympathetic Nervous System, which promotes “fight for flight” readiness.
2. Parasympathetic Nervous System, with a restorative “rest or digest” function.

In this way, the visceral perception-action cycle has two arms:

1. The autonomic nervous system, which quickly perceives and regulates the body.
2. The neuroendocrine system, which operates at a slower, more deliberate pace.

We can call the visceral perception-action cycle the “hot loop”, in contrast with the cognitive cycle, or “cold loop”. These loops represent a central organizing principle of the nervous system.

 

The spinal cord bears nerve fibers in service of both loops, both autonomic nerves bearing sympathetic & parasympathetic signals, and somatic nerves which regulate the skin and musculature. The cranial nerves include exteroceptive nerves as well (vision, hearing, smell, etc) which together encode information about the external world. 

The Folly Of Spock

As we will see later, the visceral perception-action cycle participates in the neural basis of emotion.

Next time you watch Star Trek, you may safely infer the Spock’s species has not achieved neuroendocrine integration.

Emotional processes reside at the center of the brain. Emotion is thus ideally situated to modify, modulate, and alter your decision-making capabilities.

This tension between visceral and cognitive, between hot and cold, is one of the hallmarks of being human.

Until next time.

 

Part Of: [Neuroanatomy] sequence
Next Up: The Interlocking Loop Hypothesis

Posterior Perception, Anterior Action

Consider again the neural tube. In Brain Ontogeny, we learned how brains emerge from five little bumps in the neural tube.

The neural tube develops under several organizing principles. One of the most important is the Bell-Magendie law: in the nervous system, motor nerves & computations happen in the front of an animal (anterior action), perceptual nerves & computations are at the back (posterior perception). This principle applies to the entire nervous system, from the spinal cord to the neocortex.

From an evolutionary vantage point, this principle is fairly obvious. Because gravity is a thing, the vertebrate body plan features anterior limbs (movement requires limbs between body & ground) and posterior vision (because of the sun’s position relative to the Earth). Notice how bipedalism and its deformation of the human neuraxis obscures this principle. Our quadruped ancestry thereby illuminates the functional organization of the human nervous system.

The Abstraction Hierarchy

The nervous system is a complex organ. Sometimes, choices do not require much analysis. Often, however, a decision becomes clear only after contemplation.

As information travels through the brain, it becomes more abstract, and resultant behavior more flexible. This is the brain’s abstraction hierarchy.

Darker colors denote concrete representations, lighter colors denote abstraction.

In cognitive science, the following synonyms hold:

• Information is a synonym for memory.
• Sites of information processing is a synonym for modules.

The abstraction hierarchy can inform both levels of analyses. Let’s see an example.

Cognitive scientists like to distinguish between information encoding events (episodic memory) and information encoding concepts (semantic memory). The abstraction hierarchy explains the relation between the two: the concept of apple generalizes any number of events that contain apples. That is, semantic memory is situated at a higher level than episodic memory.

How do we know the brain is organized hierarchically? Brain damage provides one line of evidence.

• Severe damage to the brain stem is fatal.
• However, if a human loses everything but the brain stem, they cling to life (persistent vegetative state)
• If the midbrain is spared, such that a person loses only their cerebrum, they can spontaneously move, but their behavior is disorganized.

Renowned neuroscientist Charles Sherrington once described the brain as an enchanted loom:

The brain is waking and with it the mind is returning. It is as if the Milky Way entered upon some cosmic dance. Swiftly the head mass becomes an enchanted loom where millions of flashing shuttles weave a dissolving pattern, always a meaningful pattern though never an abiding one; a shifting harmony of subpatterns.

The abstraction hierarchy teaches us: the higher you climb the enchanted loom, the more flexible your dance.

Biological Decision Making

We are now in a position to understand decision making.

Decisions are bridges between perception and action.

Given your brain’s abstraction hierarchy, people have invented many names for this singular concept:

Until next time.

Part of: Neuroanatomy sequence
See Also: Introduction to Homeostasis
Content Summary: 400 words, 4 min read

Homeostasis

Consider the thermostat. It consumes sensory information, and transmits electrical signals to your furnace.  In fact, one of the most important things to know about a thermostat is its input-output maps. If you know how your thermostat will behave at any temperature, you can reasonably say that you understand the device.

Suppose you return from vacation, and set your thermostat to room temperature. The thermostat turns on the furnace. How does it know when to stop?

Most thermostats accomplish this by comparing the current temperature with the desired temperature, or goal. Imagine the current temp is 60 degrees, and desired  temp is 70 degrees. The difference between the two (70-60 = 10) is called the error signal, which the thermostat uses to adjust its current strategy. If the error signal is high, the furnace might be turned on full-volume; if it is 0.5 degrees, the furnace might be turned down, to avoid overshooting the target.  We call this process negative feedback (negative, because the error signal is constructed by subtracting from the target state).

Negative feedback is the mechanism by which a thermostat to exert its “will” on the environment.

• A control system might be overdamped, and approach the goal too slowly.
• Sometimes a control system will be underdamped, and overshoot the goal before “coming back around”.
• Ideally tuned systems are critically damped, and suffer from neither of the above problems.

This ability to realize and maintain a goal state, is called homeostasis.

Cybernetics: Interdisciplinary Implications

The formal study of control theory arguably began in 1868, by physicist James Maxwell. However, control theory is relevant to many other fields, including:

• Machine Learning. Remember error signals, above? Scalar errors drive reinforcement learning, multidimensional errors drive supervised learning.
• Biology. Life is a control process whereby an organism regulates itself. Body temperature maintenance is a classic example here.

The study of biological control systems is known as cybernetics.

There exists one particular theorem within cybernetics that is of singular interest. Consider the Good Regulator Theorem:

Every effective regulator of a system must include a model of that system.

The mathematics behind this statement are explored here. But allow me to inject a biological interpretation onto this result.

To be effective at survival, every animal must have a concept of itself.

Biologically, this theorem explains why we find a cortical homunculus (“little man”) etched onto our cerebrum.

Philosophically, this theorem hints at how complex animals like us came to acquire a sense of Selfhood. These implications are explored here.

Part Of: [Neuroanatomy] sequence

Sagittal, Transverse, and Frontal

An efficient way to navigate three-dimensional spaces is to use three orthogonal axes.

• Pilots use roll, pitch, and yaw.
• Geometricians use x, y, z.
• Anatomists use sagittal, transverse, and frontal.

We also need a language to navigate anatomical space.

• Lateral/Medial means side/center (adjusting X, moving the sagittal plate left & right)
• Superior/Inferior means top/bottom (adjusting Y, moving the transverse plate up & down)
• Anterior/Posterior means belly/back (adjusting Z, moving the frontal plate forwards & backwards)

Notice how directions along the X axis (first bullet) are of a different style than other axes. While the other directions gesture at an infinite scale (-∞, +∞), the first gestures at [0, +∞).

This simplification is only possible because of symmetry: nearly all vertebrates exhibit bilateral symmetry along the sagittal axis.

Neuraxis Distortions

Anatomy maintains a lot of duplicate names (synonyms). Anatomy coordinate systems are no exception:

• Ventral/Dorsal is synonymous with front/back (Anterior/Posterior)
• Rostral/Caudal is synonymous with top/bottom (Superior/Inferior)

To be perfectly frank, anatomy naming conventions seem inefficient to me. I wonder if too much has been sacrified in pursuit of historical contiguity. But sometimes, “redundant” synonyms come in handy. Consider the following example:

Humans are bipedal. It turns out that this evolutionary innovation deforms the neuraxis.

 

Because of the bend in the human neuraxis, “front & back” is in danger of being confused with “top & bottom”.

Fortunately, we can use the above synonyms to disambiguate direction in the brain.

• In the brain alone, Ventral/Dorsal becomes synonymous with bottom/top (Inferior/Superior)
• In the brain alone, Rostral/Caudal becomes synonymous with front/back (Anterior/Posterior)

 

Perhaps humans should just give up this walking upright business altogether. It would make anatomy much easier!