Changes of physical parameters over time initially involved graphical recordings which were restricted to a single location in space, for example, plotting the changes in temperatures in a city, or the growth of an organism, or the changes in voltage of an electrical circuit. These graphic recordings of changes in time of a physical parameter at a single location in space are often called ‘traces’.
The screen shows a trace of an input encoding the Y (vertical) position with the X (horizontal) position encoding time.
From: https://en.wikipedia.org/wiki/Oscilloscope
Graphical representations of phenomena over larger spaces were developed by meteorologists plotting local values of a variable such as temperatures across a land mass. Similarly, geographers plotted the populations in cities of the world on a flat geographical-space map of two dimensions. In these cases, a third dimension (in these examples, temperature or population) is plotted against the 2-dimensional spatial map. Variation in the third dimension is often shown graphically with colour.
The strength of the wind is indicated by the colour and the direction of the wind is indicated by the arrows at each location.
(In this example, the wind data itself has 2-dimensions.)
From: http://www.bom.gov.au/australia/meteye/?loc=SA_FA001
The merging of these two graphical ways, one portraying changes over time, the other plotting values at different locations, led to the construction of spatio-temporal maps, also called ‘space-time plots’, that show how a variable changes in time over multiple locations in space. Thus, these representations are in effect 4-dimensional objects.
The underlying mathematical nature of computer-generated spatio-temporal maps allows the possibility not only to visually detect novel phenomena but also to easily quantify them. Adding further experimental values of any measurable parameter to spatio-temporal maps opens up enormous opportunities for quantitative analysis of any natural phenomena.
This conceptual frame of graphically portraying changes in natural phenomena through time and space raises the question of how to refer to specific patterns in these spatio-temporal maps that correspond to what we regard as ‘events’. The simple strategy suggested here is to extend the idea of ‘objects’, normally regarded as 3D structures, by adding in their time dimension, since everything we call ‘objects’ clearly has a beginning and presumably an end. This would apply equally to a glass, a dog, or a star. They are 4D structures with underlying dynamic processes involved in their formation and termination. Similarly, every natural phenomenon with a beginning and an end in time and space can be regarded and investigated as a 4-dimensional structure in space and time.
We cannot graphically portray four-dimensional objects as a single image or static representation. However, dynamic events graphed in spatio-temporal maps can be portrayed by sequences of 2D or 3D still images.1
Spatio-temporal Maps of the Intestine and Heart
The ideas behind the need to represent natural phenomena as four-dimensional structures arose as part of my research of the intestinal movements (such as peristalsis) which are controlled by relatively simple neural circuits within the gut wall (the Enteric Nervous System). In parallel with other colleagues, we developed methods that go beyond the recording of localised traces of movements of the intestinal wall at single sites.
From Costa and Furness, 19762.
Later, we used high-resolution videos of intestinal movements and digitally extracted the changing diameters along entire segments of gut which were portrayed as grey scale spatio-temporal maps. Suddenly, the fuller complexity of the patterns became evident, producing a more faithful representation of what was going on along the gut. These spatio-temporal maps of intestinal motility enabled us to detect visually novel patterns of movements. Being digital maps, appropriate quantitative analysis of incidence, extent, direction, and speed of intestinal contractions can be extracted from the data.
Contractions of the intestinal muscle correspond to white areas on the spatio-temporal map and upward deflections on the traces. The map contains much more data than the traces. It allows complex patterns to be visualised as well as allowing multiple samples to be analysed.
With this method, the individual events of intestinal peristalsis become recognised as discrete 4D structures with a beginning and an end. We combined spatio-temporal diameter maps with other physiological parameters including the forces responsible for changing the shape of the gut wall, allowing us to deduce the state of muscle contraction or relaxation. This combined strategy enabled us to relate the movements of the intestinal wall to the enteric neural circuits responsible for its motor control. In doing so, we identified several novel intestinal motor patterns, significantly advancing the field of neurogastroenterology3.
In the heart, a pacemaker system of coupled cells oscillate and generate the rhythmic heartbeat. Unlike the intestine, the presence of specific ‘nodes’ of higher frequency pacemaker cells guarantees that the rest of the heart cells are driven with the sequential synchrony required for effective mechanical pumping of the blood. When the nodes no longer drive the heart in synchrony, the heart enters into fibrillation. The patterns of electrical activation in this condition have been constructed graphically and analysed as spatiotemporal maps.
From: Strocchi et al, 2020.4
The ability to represent graphically natural phenomena as 4D structures gives us a most powerful tool to relate visual patterns with scientific observations. We can now apply this method to neural events and their subjective counterparts. This process requires us to regard mental phenomena as natural phenomena which can also be represented as 4D structures.
I will argue in the remaining of this essay that constructing spatio-temporal maps of brain activity as 4D structures will provide the physical bases of neural phenomena starting from simple locomotion up to complex mental states and will help bridge the apparent gap between subjective and objective experiences.
- Video animations allow a more direct 4D visualisation of spatio-temporal maps changing over time. ↩︎
- M Costa & JB Furness (1976): The peristaltic reflex: an analysis of the nerve pathways and their pharmacology. Naunyn-Schmiedeberg’s Archives of Pharmacology 294, 47-60. ↩︎
- For examples, see: M Costa, TJ Hibberd, LJ Keightley, L Wiklendt, JW Arkwright, PG Dinning, SJH Brookes & NJ Spencer (2019): Neural motor complexes propagate continuously along the full length of mouse small intestine and colon. American Journal of Physiology 318(1) G99-G108;
M Costa, LJ Keightley, L Wiklendt, TJ Hibberd, JW Arkwright, T Omari, DA Wattchow, V Zagorodnyuk, SJH Brookes, PG Dinning & N J Spencer (2019): Roles of three distinct neurogenic motor patterns during pellet propulsion in guinea pig distal colon. Journal of Physiology 597(20), 5125–5140;
M Costa, TJ Hibberd, L Keightley, L Wiklendt, MA Kyloh, P Dinning, SJH Brookes & NJ Spencer (2021) Novel intrinsic neurogenic and myogenic mechanisms underlying formation of faecal pellets along the large intestine of guinea-pigs. Journal of Physiology 599(20), 4561-4579;
M Costa, L Wiklendt, T Hibberd, P Dinning, N Spencer & S Brookes (2022): Analysis of intestinal movements with spatiotemporal maps; beyond anatomy and physiology. The Enteric Nervous System II: Advances in Neurogastronterology 1383: 271-294. ↩︎ - Strocchi M et al (2020): A publicly available virtual cohort of four-chamber heart meshes for cardiac electro-mechanics simulations. PLOS One 15(6): e0235145;
For more examples, see: KHWJ Ten Tusscher, R Hren & AV Panfilov (2007):Organization of Ventricular Fibrillation in the Human Heart. Circulation Research (100) e87–e101;
J Jalife (2003): Rotors and Spiral Waves in Atrial Fibrillation. Journal of Cardiovascular Electrophysiology 14(7) 776-780.
RA Gray, AM Pertsov & J Jalife (1998): Spatial and temporal organization during cardiac fibrillation. Nature 392(6671) 75-78. ↩︎. ↩︎