Abstract
Cognitive neuroscientists are turning to an increasingly rich array of neurodynamical systems to explain mental phenomena. In these explanations, cognitive capacities are decomposed into a set of functions, each of which is described mathematically, and then these descriptions are mapped on to corresponding mathematical descriptions of the dynamics of neural systems. In this paper, I outline a novel explanatory schema based on these explanations. I then argue that these explanations present a novel type of dynamicism for the philosophy of mind and neuroscience, componential dynamicism, that focuses on the parts of cognitive systems that fill certain functional roles in producing cognitive phenomena.
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Adapted from Kelso (1995, p. 57)
Adapted from Roitman and Shadlen (2002, p. 9482)
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Notes
For this essay, I adopt an object, property, and relation ontology. On such a view, objects are analyzed as bundles of properties with some substrate. This approach is adopted for convenience only, and the claims can be suitably recast in other terms. In order to cast a wide net, in this section I have framed the view in terms of individuals in the logical sense (cf. Strawson 1959). I thank a reviewer for pushing me on this point.
This distinction between dynamical systems and dynamical systems theory is often overlooked. I explore and defend this distinction elsewhere.
Systems can also evolve with respect to other variables besides time as noted, but to simplify I will often focus on time.
Some dynamicists might object to such a characterization of dynamical systems as ignoring parts of these systems. Specifically, Bechtel classifies connectionist models that refer to internal operations as dynamical systems that do refer to parts (Bechtel 1998). Bechtel says that “the question of how [dynamical] models comport with mechanistic explanatory objectives does not arise, since connectionist models are models of mechanisms. The [dynamicist] approach is employed by these theorists to analyze how these mechanisms behave” (Bechtel 1998, p. 312). This analysis involves decomposing the mechanism into parts and localizing the operations of the mechanism to the activity of these parts (Bechtel 1998; Bechtel and Richardson 2010). However, not all philosophers agree with this broad characterization of connectionism’s relation to dynamicism. For example, Zednik disagrees, arguing that “[i]nsofar as connectionist dynamical models do not… capture the dynamics of coupled brain-body-environment systems, but merely describe “internal” neural dynamics, they differ substantially” from other connectionist models that focus on “the behavior of the whole… brain-body-environment system” (Zednik 2008, pp. 1457–1458). These latter systems “can be understood as the interactive operations between two simpler subsystems: the agent’s motion M as the operation of subsystem A (the agent) on the one hand, and the sensory stimulus function S as the operation of subsystem E (the environment) on the other” (Zednik 2008, p. 1458). Notably, the behavior of these models is explained by an “understanding of the way in which individual parts of the system—the agent on the one hand and the environment on the other—interact” (Zednik 2009, p. 2301). There is thus some debate as to the appropriate interpretation of certain connectionist systems, and some such systems do refer to parts or parts of parts while others do not. My componential dynamicism is consistent with connectionist approaches that focus on internal dynamics.
Kelso himself would probably endorse parts, seeing as his research program includes both understanding how the neural mechanisms give rise to the dynamics captured by the HKB model and applying the lessons gleaned from dynamical analysis of behavioral phenomena to the brain. I do not have space to fully address Kelso’s own dynamicism (Kelso 1995), which is a remarkably deep and distinct application of dynamical ideas to analyzing cognitive phenomena and which should also be seen as an exception to the systemic approach. Kelso’s view is that there are dynamical laws that underlie many complex systems, including cognitive ones, and that cognitive phenomena result from these laws. In addition to this nomological component, his view has a mechanistic component, seeing the dynamical systems as mechanistic parts. For the time being, suffice to say that Kelso is not a target of my criticism.
From now on, I will exclusively use the term ‘parts’ to refer to subsubsystems of the system.
See below, “Functional neurodynamics” section, for a lengthier discussion of functions and componential dynamical systems.
Note that the parts of the fingers like their nerves or muscles do not necessarily determine the relative phase. Usually, those properties will determine the fingers’ movements. But there could be ways of setting the value for the relative phase other than the musculature, nerves, and other physiological parts that typically cause finger movements. Imagine, for example, that a mechanical device is used to oscillate the fingers, mimicking the behavioral output of the usual physical parts. On systemic dynamicism, this mechanical system is described in the same way as when the musculature and nerves control the movements. Systemic dynamicism cannot distinguish between the two systems, one organically controlled, the other artificially, because the role of the parts is excluded by systemic dynamicism. The inclusion of a function describing the motive forces on the fingers could distinguish between different sources of movement. Such a description will distinguish different sources of movement unless the proximal causes of the finger movements—viz., muscles and nerves versus artificial means—have precisely the same force functions in all contexts. But even then, the equations corresponding to such functions are interpreted; so the variables refer to those proximal causes, which will be organic in the first case and artificial in the second.
This division of types of dynamical systems is not meant to be a stark contrast. There can be systems with mixed dynamical descriptions. A car can be described systemically, componentially, or in a mixed fashion, with some terms for its environment and systemic properties and other terms for the properties of the parts of the car and environment. Thus, the two types of dynamical description stand at extremes on a scale of dynamical descriptions, with most such systems lying somewhere in the midst of the spectrum.
From this point on, I will focus on changes in properties, but this should not be read as excluding changes in objects or relations as underlying dynamical properties.
In fact, neurophysiological systems implement dynamical systems in a technical sense that I explore in other work. For the time being, instantiation here should be understood in the sense of token identity, so that if a neuron instantiates a dynamical system, then a subset of the neuron’s dynamical properties are an instance of the dynamical system; if a neural population instantiates a dynamical system, then a subset of the population’s dynamical properties are an instance of the dynamical system; etc. In addition, the neuron, neural population, etc. are themselves dynamical systems.
See Gold and Shadlen (2007) for extensive discussion of this research. Note that many aspects of this case are still actively researched. In particular, whether neurons in area LIP truly exhibit the integrate-to-bound dynamics discussed below is hotly debated (Latimer et al. 2015). For my present purposes, the fact that the details are still not settled does not matter, as I am merely illustrating how such explanations are constructed.
When describing this state change mathematically, both the smooth and non-saltatory nature of the change is captured by a continuous mathematical function. The mathematical function does not contain jump or point discontinuities, corresponding to a lack of saltatory changes in the system. The function is also continuously differentiable, corresponding to a lack of abrupt changes in the system. While there is also typically some monotonicity in the trajectories, the trajectory’s evolution is correlated with some input such that if the input changes sign, then the trajectory can switch from an increase to a decrease or vice versa. A description of a system’s evolution wherein the system resides in the same state despite changes in some variable does not count as an example of integration.
In other work, I intend to present a deeper analysis of functions.
Causal functions may be clearest example of compositional functions. Constitution is taken to be distinct from causation for a number of reasons, including that the former is synchronic and the latter diachronic. Whether a component of a physical system can help constitute that system without causal powers is an interesting open question, though this has recently been argued by a range of philosophers (see e.g. Huneman 2017 or Chirimuuta 2017).
Here, I deliberately include both causal and constitutive roles for the part in c-functions.
This model is often dubbed the drift diffusion model for historical reasons.
The system could also run out of evidence (e.g., if the dot display disappears) or some other stopping criterion could be met that results in a decision.
Carandini and Wang both discuss the neurons as computing an exponentiation function (Carandini and Heeger 2012; Wang 2002). In a case of pure exponential growth, for a given strength of evidence, the relative change in the firing rate, or the change in firing rate divided by the current rate, is constant (Stewart 2011). However, upon removal of the stimulus, LIP neurons will not maintain their firing rates indefinitely, and adding state-dependent leak and recurrent excitation exponential decay terms captures this decay (Usher and McClelland 2001; Wang 2002). Furthermore, the integration is noisy, requiring the addition of a noise term. Thus, neurons such as those in area LIP are ‘noisy leaky integrators’; their firing rate is a function not only of the change in firing but also contains a noise term and a loss (or ‘leak’) term (e.g. see Usher and McClelland 2001 or Wang 2002; for a philosophical discussion, see Eliasmith 2013, p. 43ff). The leak is dependent upon the state of the system, and so the internal system dynamics contribute to the state of the system. Wang’s model is even more complex but in short contains an additional recurrent excitation term that is driven by the state of the system. These noise and leak terms effectively curtail the integrative growth function as well as enforcing a biologically plausible decay in the firing rate at some time constant absent any driving force, while the differences in the strength of evidence correspond to differences in the relative growth rate constant driving the neuron’s activity.
Exactly how many steps depends on the number of subcapacities of the capacity. The set of subcapacities often implies some ordering to their execution. Some subsets of subcapacities may be executed serially or in parallel, whereas other subsets must be executed seriatim.
Does the part need to be causally active in the cognitive system? I will remain neutral on this issue herein; some explanations of cognitive capacities might reflect purely structural features of systems such that the parts’ causal powers are not relevant (see e.g. Chirimuuta 2017 or Huneman 2017). In the examples herein, causality is important, but I do not want to beg any questions about other cognitive phenomena.
For this discussion, the part of the brain, area LIP, is a subsubsystem because the subsystem is the brain itself.
Perhaps these processes should not be considered cognitive. I construe the term ‘cognitive’ quite widely, so I set aside this concern.
The computations executed by these neural dynamics are often magnitude to magnitude transformations or functions on magnitudes. I note that these computations are not obviously like the representational transformations posited by classical computational theories like the language of thought (Fodor 1975).
Hence, showing that e.g. a generalized linear model containing some independent variable can significantly capture some of the variance in neuronal firing rates is insufficient to show that the neuron’s function is to signal that variable. In order to argue that the neuron’s function is in part to signal the variable, there must be at least the additional step of modeling the neuronal dynamics, such as by fitting a Gaussian tuning curve.
Despite instantiating the same dynamical system, the physiological systems are different in LIP and ACC. In the case of LIP and the RDMT, integrative activity occurring on the timescale of hundreds of milliseconds during a single trial is instantiated by an increase in firing by individual neurons. In the case of ACC and the patch foraging task, integrative activity occurring on the timescale of tens of seconds over many trials is instantiated by an increase in the peak activity of individual neurons during a trial. These distinct timecourses indicate distinct physiological mechanisms instantiating the integrate-to-bound system. Furthermore, the integrate-to-bound in LIP is instantiated via a noisy but continuous-like increase in firing rates, whereas the integrate-to-bound in ACC is instantiated in a series of discontinuous transient increases in firing rates. These distinct firing rate patterns also indicate distinct physiological mechanisms. These two differences suggest that distinguishing between the dynamical system and the physical mechanism that possesses the dynamics is necessary to understand how cognitive neurobiologists go about explaining cognitive phenomena. In addition, the differences in the integrative activity will be reflected in models that include greater neuronal details. More abstract models, however, that focus just on the integrating peaks of activity may still use the same mathematical operations as used to describe the integrate-to-bound activity in area LIP.
This will also illustrate how the schema applies well beyond decision making to a range of cognitive capacities.
I cannot do justice to the two decades long research project, and the dozens of studies, that have gone into the development of the DN model of attention. An excellent review is in Reynolds and Heeger (2009).
The basic mathematical description of divisive normalization is
\( R = \gamma \frac{{D_{j}^{n} }}{{\sigma^{n} + \mathop \sum \nolimits_{k} D_{k}^{n} }} \)
for response of a cell R, input to the jth cell Dnj, and normalization pool summed over the normalization input Dnk (Carandini and Heeger 2012, p. 54). The parameters γ, n, and σ are fit to the data, with σ controlling how quickly the firing of a cell reaches its maximum. When σ is very large, the normalizing input has little effect on the firing rate, and when σ ~ 0, the input is largely determined by the normalizing pool of activity. If σ is roughly equivalent to the normalization input Dnk, the cell is only moderately determined by the activity of the normalization pool.
Some models do contain such parameterizations. What’s key here is that such models need not; and, as the illustration above showed, certain central cases of explanation of cognitive phenomena by neurodynamics do not.
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Acknowledgements
This work has benefitted greatly from a number of comments over many years. The work draws on my dissertation at Duke University, and I would like to thank Walter Sinnott-Armstrong, Alex Rosenberg, Karen Neander, Felipe De Brigard, and Michael Platt for their thoughtful input. I would also like to thank Gaultiero Piccinini, Chris Peacocke, William Lycan, Ann-Sophie Barwich, Jorge Morales, Nemira Gasiunas, Michael Weisberg, and the University of Pennsylvania philosophy of science reading group for critical comments.
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Barack, D.L. Mental machines. Biol Philos 34, 63 (2019). https://doi.org/10.1007/s10539-019-9719-6
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DOI: https://doi.org/10.1007/s10539-019-9719-6