Imagine the brain as a wild ocean, with waves constantly crashing and rippling. Well, scientists have been busy studying these brain waves (with a lot fewer seagulls!) and they’ve discovered something fascinating. By applying a mathematical model called a fractional-order neural field model, they found that the fractional-order dynamics, which represent memory and hereditary properties of the brain system, play a crucial role in shaping cortical wave features observed before seizures stop. To investigate these dynamics, the researchers used two different methods: one that focused on the initial wave behavior and another that examined long-term wave propagation. Interestingly, they observed distinct effects of fractional-order on wave speed and wave width depending on whether the order was less than 1 or greater than 1. This means that different types of waves could be modeled depending on the values of the fractional order. They also found that wave width was influenced by factors such as synaptic threshold and synaptic connectivity. It’s like discovering how different factors like wind direction and ocean currents shape the waves in our brain sea! This research opens up new possibilities for understanding and manipulating brain activity in conditions like epilepsy. Get ready to dive into this scientific exploration by checking out the full research article!

In this work, we establish a fractional-order neural field mathematical model with Caputo’s fractional derivative temporal order α considering 0 < α < 2, to analyze the effect of fractional-order on cortical wave features observed preceding seizure termination. The importance of this incorporation relies on the theoretical framework established by fractional-order derivatives in which memory and hereditary properties of a system are considered. Employing Mittag-Leffler functions, we first obtain approximate fractional-order solutions that provide information about the initial wave dynamics in a fractional-order frame. We then consider the Adomian decomposition method to approximate pulse solutions in a wider range of orders and longer times. The former approach establishes a direct way to investigate the initial relationships between fractional-order and wave features, such as wave speed and wave width. In contrast, the latter approach displays wave propagation dynamics in different fractional orders for longer times. Using the previous two approaches, we establish approximate wave solutions with characteristics consistent with in vivo cortical waves preceding seizure termination. In our analysis, we find consistent differences in the initial effect of the fractional-order on the features of wave speed and wave width, depending on whether α <1 or α>1. Both cases can model the shape of cortical wave propagation for different fractional-orders at the cost of modifying the wave speed. Our results also show that the effect of fractional-order on wave width depends on the synaptic threshold and the synaptic connectivity extent. Fractional-order derivatives have been interpreted as the memory trace of the system. This property and the results of our analysis suggest that fractional-order derivatives and neuronal collective memory modify cortical wave features.

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