Mechanical forces influence the nervous system across both temporal and spatial scales. At longer time scales, mechanical forces govern folding patterns during brain development [1], [2], and at shorter time scales, forces applied at high rates cause traumatic brain injury [3]. Though mechanical forces and their effects are more easily observed at the larger spatial scale, brain folding and other larger-scale phenomena are intimately connected to cell-level responses and behaviors. For example, mechanical forces promote axon growth during development [4], and studies suggest that axon growth could play a role in directing brain folding [5]. Similarly, the macroscale forces involved in traumatic brain injuries cause microscale damage in individual axons [6] (Fig. 1).

Within the axon, the cytoskeleton acts to both generate forces and to provide structural support. Microtubules run discontinuously along the length of the axon, and crosslinking proteins bundle these microtubules together to create the core of the axonal cytoskeleton [7]. While passive crosslinks like tau contribute mechanical support in response to external loading, the motion of active crosslinks like dynein generates active forces within the axon [8]. Neurofilaments, another major component of the cytoskeleton, form an extensive network and regulate the axon diameter [9]. Surrounding the microtubules and neurofilaments, spectrin alternates with actin rings to compose the actin cortex [10]. Actomyosin contraction within the cortex supplies another source of active force generation [11]. These various cytoskeletal elements cooperate in a delicate balance of forces to maintain the structural integrity and biological function of the axon (Fig. 2).

In relating subcellular phenomena to larger scale behaviors, computational models work together with experimental studies to provide additional insight. Analytical and numerical models allow scientists to probe the isolated effects of various cytoskeletal parameters that are difficult to study using experimental methods alone. In axon growth, experimental studies have highlighted several sources of mechanical force generation [12], [13], [14], and computational models have investigated how these forces might interact to generate emergent behaviors of axon elongation and contraction [15], [16], [17]. Similarly in traumatic brain injury, experimental studies have discovered evidence of subcellular damage [18], [19], [20], and models have studied how mechanical loads of different magnitudes and rates could cause the observed damage [21], [22]. Insight from these computational studies guides the development of new experimental approaches and predicts important areas for future research.

In this review, we discuss the use of mathematical models in studying axon mechanics. Starting at longer time scales, we present models related to understanding the role of mechanical forces in axon development. Next, we discuss models studying the use of tension to promote axon growth and regeneration. Finally, we examine injury-level loads and models of axon damage.