The stochastics of divergent thinking

(This is the third blog post in a series on the stochastics of human and artificial creativity. You can read the first two posts here and here)

Creativity is a random process that resides along the edge between chaos and control. This was the conclusion of the previous blog post, where we explored the ideas of David Bohm. In this blog post we will look closer into the chaotic, or stochastic, part of creativity.  Divergent thinking or thinking outside the box is often used to describe this somewhat unpredictable exploration of the unknown in the pursuit of potential new, creative ideas. If we seek to understand this random component of human creativity, a closer look into the statistics field of stochastic processes may be beneficial. This is also necessary if we want to follow up on the conjecture of McCarthy and colleagues from 1956 (see first blog post) that human creativity may, at least, be imitated by computers. 

Maybe you are familiar with the possibility to use statistics software to draw numbers as a random sample from some probability distribution, for instance from a normal distribution. There are numerous methods that are developed for such random sampling, and one group of methods is known as Markov Chain Monte Carlo methods (MCMC). These are commonly used in Bayesian modeling in statistical inference. They are computer intensive methods where a so-called random walk process is initiated in order to simulate a dependent chain of observations from a given probability distribution, often called a target distribution. In a typical set up of MCMC each successive step of the Markov chain depends on the previous through a current value, being either a starting point or the latest sampled value. Given the present, as a current value, the method contains two parts:

  1. A new candidate value, typically depending on the current, is drawn from a proposal distribution (a random step).
  2. The candidate is accepted or rejected as a new value of the chain in light of the target distribution (an evaluation step).
Stochastic processes

You may have seen the relevance to creative thinking already. This Markov chain resembles an associative chain of thoughts. The proposal distribution is at large responsible for the randomness of the creative process, whereas the target distribution (or we could say the bias) evaluates the usefulness. This is a statistical framework for setting up association processes which satisfy both requirements of a creative process, as discussed in the previous blog post in this series. What’s needed is a sensible way to generate candidate ideas and a way to evaluate the usefulness of such ideas. 

The cognitive analogy of MCMC is perhaps that of thinking inside the box. It is like a situation where we sit and reflect, making a chain of associations where all thoughts are evaluated within the same measure of usefulness, which is our current personal cognitive bias. Usually  we speak of creativity as thinking outside the box, though, but as we will see later, thinking outside the box is all about challenging our old boxes, that is, by bending, blending or breaking the biases we are using to judge our thoughts by. I will return to the importance of biases in the next blog post. Now I will use the MCMC example and the statistical framework of Markov models to dig deeper into the stochastic properties of creativity.

Introverted thinking is a fully associative process. One thought leads to another. If we want to recall memories, we cannot look them up like we use the index pages of a dictionary. We need to get hold of them through associations. In neuroscience a common understanding is that thoughts and memories are stored in long-term memory as paths of strongly connected neurons, sometimes called «engrams», organized in a huge association network.

The psychologist Donald Hebb described learning at the neuronal level (in 1949) as: “neurons that fire together, wire together”. The Hebbian theory describes how synapses, the neuronal connections, are strengthened in a learning process if two connected neurons are activated simultaneously or successively during experiencing or learning. However, the reverse phrase, “neurons that wire together, fire together”, is perhaps more appropriate to describe the replay processes we perform whenever we are recalling memories and learned facts. Random signal processes will tend to follow well established routes, either as a conscious chain of thoughts, or in the realm of our unconsciousness. In fact, chains of connected memories are replayed even when we sleep, as found in rodents by Wilson and McNoughton (1994)

Let’s pause a bit and check out some of your wirings. What is your first association with the word ‘yellow’? Maybe you were thinking ‘sun’, ‘banana’ or perhaps ‘submarine’, whereas ‘car’ would probably be less likely (unless you happen to own a yellow car). The probabilities of moving to other thoughts from the current thought would be called transition probabilities in a statistical language. My personal transition probability from ‘yellow’ to ‘submarine’ is relatively high since I remember well the Beatles’ hit ‘Yellow Submarine’. After thinking ‘submarine’ my next immediate associations were ‘Beatles – Lennon – Shot – Dead’. Your association chain would probably follow another route of random walk, and so would perhaps mine in another circumstance.

Let’s first assume that at this very moment your thoughts are sampled from a fixed repertoire of potential thoughts and memories. We may call the thought repertoire of yours the state space of thought. The probability of the various cognitive outcomes from this state space depends on your entire history of experiences, and together they define the distribution of transition probabilities over your personal state space of thought. For simplicity we first assume that the transition probabilities do not change over time. Some transitions are highly probable, others are quite unlikely. From this distribution you make a random choice of what to think next. As seen in our little statistical example above, Markov processes (named after the Russian mathematician Andrey Markov) can to some extent approximate cognitive processes. Markov processes have many interesting properties which are cognitively relevant also for creativity. I will highlight some of them here.

The primer effect

The first property is the ‘primer effect’. A random walk process must start somewhere, hence it requires a starting value, or a so-called primer. From the primer the process ‘walks’ from one value to another. Accordingly, for a cognitive process the current thought is very often the primer for the next thought. 

The effect of priming is well known and studied through many psychological studies. Priming describes how thoughts and decisions may be biased towards input cues. The cues may be more or less random or given deliberately to manipulate the cognitive response of the receiver of the cues. Priming is, for instance, a widely used technique in consumer marketing where subtle messages are given to influence our opinions about products to increase the likelihood of us buying them. Another example is social media marketing giving highly personal primers based on the information we provide online. 

The priming effect is also important in the context of creativity. In the absence of external stimuli, chains of thoughts are likely to be rather introverted and mainly traverse highly familiar ground in our state space of thought. In a sense we are then not primed to be novel.  Larger jumps may be primed if we are open to our surroundings or perhaps even seek new environments and perspectives, for instance by traveling to explore new cultures, taking another route to work, or reading a random book from the library. This may trigger chains of thoughts that explore paths less travelled by in your state space of thought. Kounios and Beeman discuss multiple factors in their book “The Eureka factor: Aha Moments, Creative Insight, and the Brain” that can prime insightfulness.

Focus level

Another important parameter in random walk processes is the step length of the walk. This is typically controlled by the proposal distribution. We can think of this as the focus parameter of the process. If step lengths are short, the process moves very slowly across the state space, only entering closely connected states within long time spans. Furthermore, the series of visited states will show a high level of so-called auto-correlation, which in the cognitive setting means that thoughts tend to be similar and related over time. One might characterize a person with highly auto-correlated thinking as narrow-minded. Short steps plus intentional priming is the way recommendation services like Netflix or social media feeds manage to keep you continuously satisfied. Unfortunately, it may also have the effect of limiting the range of impulses and perspectives. However, being narrow-minded is also necessary, for instance when we have to focus strongly on solving some difficult task or concentrate on learning some new skill. 

Neurologically, strong focus is induced in the brain by activation of inhibitory neurons through increased release of the neurotransmitter GABA (Gamma-aminobutyric acid), which reduces transition probabilities for long step transitions to seemingly irrelevant or distracting thoughts. GABA has the effect of shutting out external and distracting signals. This is the case during stages of fancy, as discussed by David Bohm, especially during what he referred to as rational fancy (see previous blog post in this series). If step lengths are allowed to increase (by a reduction of GABA, focus is lowered and a more diffuse state of mind is induced, as in Bohm’s stages of insight. 

The problem with the slowly moving cognitive chain of the focused mind is the high probability of missing out on creative candidate solutions to problems. It simply takes too long to traverse the relevant areas of the state space of thought! On the other hand, too long steps may increase the risk of very remote ideas to pop up, only to be rejected as irrelevant in the current context. In the case of long step lengths, auto-correlation may be very low, and thoughts appear to be disconnected. In a previous blog post Be creative – use your noisy brain I have written about how spontaneous firing in the resting brain can be a source of creativity by nurturing long and disconnected jumps in the state space of thought. This noisy feature of the brain is a powerful source to divergent thinking.

Cognitive flexibility

In Bayesian inference so-called well-mixing Markov processes are desirable, wherein the chain moves across the state space with optimal step lengths, avoiding both being too narrow-minded and too diffuse. Such well-mixing processes have the largest probability of covering a relevant state space in sufficient proportions within a limited time span. A special kind of Markov chain is the time-inhomogeneous chain, where step-lengths (focus) are allowed to vary over time to optimize chain mixing. 

For cognitive processes the definition of good mixing will, of course, depend on the context, whether focusing or defocusing is most beneficial. Later we will see that the ability to be flexible and switch easily between short and long transitions may be beneficial for the creative process. When David Bohm in his book On creativity describes the creative process as an iterative process between stages of insight and stages of fancy, he really describes a time-inhomogeneous stochastic process of thinking. This flexibility is also a key component in a popular work mode in design and innovation processes known as design thinking.

Plasticity and learning

Thus far we have for simplicity assumed that there is a fixed probability distribution across the state space of thought for an individual, and that the state space itself is static. This is characteristic for a stationary distribution in stochastic process theory. There is, however, no reason to believe that the cognitive state space is static, nor that the thought-distribution is stationary. This is due to the fact that we are all expanding and altering the state space through learning, and the brain is continuously changing, both functionally and structurally. The associative chains of thoughts are actually changing the probability distribution over the state space as it moves. This is because repetitively running chains of association is a key part of learning, and the recall process changes the synaptic strength between the neurons. The very fact that the process visits certain thoughts or memories, increases the probability of a later revisit. Furthermore, new and previously unvisited thoughts occur during the random walk as a result of creative thinking or learning from external input. On the other hand, parts of the state space may also be almost or entirely eliminated (forgotten) through loss of synaptic connections, for instance, by rarely being revisited or by brain lesions. The brain is very plastic and changeable, and hence, so is the state space of thought. Therefore, the random walk of learning and creative thinking may be considered as a non-stationary stochastic process. If you think about it, this should be obvious. During our lifetime interest, values and the context we’re part of change, and this certainly reflects our thought processes. It is reasonable to assume that divergent thinking, seeking new perspectives and expanding one’s field of knowledge has a self-enhancing effect on creativity due to the inhomogeneity and non-stationarity properties of the associative thinking process.


Multiple Markov chains running in parallel may cover a state space faster, and integrating the information from many chains may be beneficial for many purposes in a statistical setting. Parallel cognitive processes are also integrated in the context of a conversation, for instance, at the lunch table, by the coffee machine, or during group based learning. Different people have different experiences and knowledge levels, fields of interests, values and personalities. Each person even has his or her individual cognitive state space and transition distribution. Through dialogue, parallel random walks of thought may evolve jointly towards a better understanding of some subject to be learned. 

A dialogue
A dialogue – Credits: Håkon Sparre/NMBU

However, in a dialogue, the thought processes themselves are actually not interacting, because the participants do not observe one another’s thought processes. A special kind of Markov processes, called Hidden Markov Models (HMMs), can give insight into the dialogue in that respect. In statistical inference HMMs are used to model processes where it is reasonable to assume that there is an underlying, hidden process that occasionally gives rise to observable output. Our cognitively relevant example of a HMM is your indirect observation of my thought process. My conscious thoughts are available to me, but only occasionally and approximately observable to you, and that is whenever I orally express my thoughts. 

In the brain the process of transferring conscious thoughts from the prefrontal cortex to the cortical areas responsible for speech (Broca’s area), is itself a stochastic process that adds noise to the output. This manifests itself in the fact that sometimes it is difficult to express exactly what you are thinking. HMM’s are similarly defined not only by transition probabilities for the hidden state space, but also state-dependent probabilities for generating observable and noisy output. Hence, some thoughts are more likely to be expressed than others, and with higher or lower accuracy. Furthermore, it is reasonable to assume that the output probabilities are inhomogeneous, meaning that various contextual factors and conditions have an influence on the probability that thoughts are articulated. The output probabilities may, for instance,  depend on the context of the conversation, or the personalities of the participants of a dialogue. For instance, if I am on non-familiar ground, either literary or cognitively, I am less likely to express my thoughts. Furthermore, I am an introverted person who typically is less expressive than my extraverted peers in dialogues. The cognitive processes of an introvert are generally more hidden, having smaller probabilities of generating output than the case is for an extravert. For others, my expressed thoughts as an introvert may therefore seem less connected (having lower auto-correlation) compared to an extraverted and more talkative person.

Creative team processes, involving dialogue and conversations, may be explored further in the context of parallel HMMs, but a main point in this context is that through conversation, associations are exchanged, which may lead to jumps in the thought processes of the participants. These jumps can result in a better coverage of the state space and faster idea generation and learning for each individual group member. The integration of information from multiple parallel processes becomes most effective through open and empathetic dialogue where processes have equal opportunity to influence one another. Once again David Bohm is relevant. The open, non-judgemental dialogue as an arena for creativity, is well explored in his book On dialogue.

The powerful dialogue of the unconscious brain

Parallel associative processes also occur in another cognitive domain, namely in the unconscious mind. The brain is never at rest. Even when we have dreamless sleep, the neurons fire and pass signals to each other. The aforementioned replay of thoughts along probable neuronal paths, run in our unconsciousness, visiting thoughts, memories and ideas, and maybe they are related to some unresolved issues or problems you have been focusing on lately. This resembles divergent conscious thinking, but recent studies indicate that the unconscious is much more powerful than the conscious mind as a provider of novel thoughts and ideas. There are several properties that make the unconscious so effective for divergent thinking, for instance:

  • Parallel processes  As discussed above under dialogue, parallel neuronal processes may cover a state space of thought faster than a single (conscious) process. Since the unconscious mind is not restricted to give attention to a single thought process, like the conscious mind is, it is reasonable to believe that there are multiple parallel cognitive processes running beneath the surface of awareness. The psychologist and Nobel laureate David Kahnman gives support to this in his book Thinking Fast and Slow where he describes the fast thinking, automated, and unconscious ‘system 1’ in the brain as a parallel processing system. The slow thinking and conscious ‘system 2’, he states, is a serial system. This multivariate thinking process of the unconscious mind may be highly beneficial if it is coupled with some information integration system. We will return to the latter aspect later.
  • Reduced bias  As I will return to in the next blog post, our biases may direct influence on the random walk process of divergent, conscious thinking, but it is reasonable to believe that biases are less influential on unconscious thoughts. I guess most of us have experienced divergent thinking during dream sleep which is far beyond what we perhaps would allow ourselves when we are awake. The famous psychiatrist and psychologist, Milton Erickson, who specialized in medical hypnosis, said that biases are the province of the conscious mind, and that the unconscious level of awareness is characterized by the absence of the influence of biases. Further, he declared that “the unconscious typically involves a more objective and less distorted awareness of reality than the conscious”.
  • Spontaneous firing  As mentioned above under the focus level property of stochastic processes, spontaneous firing may cause longer jumps between conscious thoughts, especially when we lower our focus level. For the unconscious brain, it is reasonable to believe that spontaneous firing may be an even richer source to divergent thinking, since the unconscious is likely to be less affected by the focus level and the biases of the conscious mind.
  • Increased speed  Experiments with rodents have shown that the speed of the replay of memories during sleep increases multiple times. It appears that the unconscious thought processes may be accelerated in time during so-called sharp-wave ripples going on in the hippocampus (Liu et al,,2019). These are high frequent bursts of waves travelling through the neural network of the hippocampus and cortical regions. Ivry (1997) discusses the role of the cerebellum as a timing device which may control the speed of the replay. This small, but complex brain region is known to play an important role in controlling and coordinating bodily motor skills, but recent research points to the possibility that the cerebellum also plays an important role in creative processes.  I will return to this in the next blog post.
  • Reshuffling and randomization  A recent study by Stella et al. (2019) on rats indicates that thoughts and memories are not only replayed during sleep, but sequences of memories may be randomized during replay. Liu et al (2019) also found that humans unconsciously replay, reshuffle and reorganize experiences. This randomized replay may be an important source to unconscious divergent thinking where new combinations of thoughts and ideas are tested.

So we might say that the brain is running an open and objective dialogue, or even a multilogue, with itself below the surface of consciousness. Probably this also is as close as we can get to an internal Bohm Dialogue. The philosopher and scientist David Bohm was a strong advocate for the open dialogue as a playground for creativity.  In this way your unconscious brain is a highly advanced and multivariate statistical processor that may provide a wide variety of candidate thoughts and ideas through recombination and reshuffling of learned sequences, and at high speed. The only drawback is that most of this dialogue happens beyond our reach, in the unconscious. Nevertheless, every now and then this powerful statistical machinery serves some insight, an eureka moment, to the surface of attention. This is the imaginative insight described by Bohm. However, in order to do so, the unconscious can not only bring up candidate ideas, the unconscious must also evaluate, at least to some extent, the usefulness of these candidates before any insight unfolds before the mind’s eye. It is here that our biases, internal models and structures come into play. 

The stochastic process properties discussed here are highly relevant for implementing virtual association processes on the path towards AC. Inspiration from human divergent thinking, both conscious (serial) and unconscious (parallel) could lead to efficient and fast idea generation. With regard to the conjecture from the Dartmouth proposal stating that human creativity should be possible to imitate in silico, this is perhaps the easy part. Computers are really good at providing random (or pseudo-random at least) numbers as a basis for divergent stochastic processes. The tricky part of creativity simulation will be covered in the next blog post where we will look into the important role of biases, how they work, are created, maintained and often dissolved throughout lifelong learning, and, of course, their important role in creative processes.

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